GCSE Maths: Pythagoras’ theorem: finding the hypotenuse and a shorter side (Foundation)
Part of the GCSE Maths pythagoras and trigonometry chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Pythagoras’ theorem: finding the hypotenuse and a shorter side
A right-angled triangle has one angle of 90° (a square corner). The longest side of a right-angled triangle is the one opposite the right angle. It has its own name: the hypotenuse. The other two sides are called the shorter sides (or the legs).
More than 2500 years ago, people noticed something special. Draw a square on each side of a right-angled triangle. Then the area of the square on the hypotenuse is exactly equal to the areas of the two other squares added together. In the picture below, the squares on the shorter sides have areas 9 and 16, and 9 + 16 = 25, which is the area of the biggest square.
This is Pythagoras’ theorem. If the shorter sides are a and b and the hypotenuse is c, then
a2 + b2 = c2
Remember from Chapter 2 that a2 means a × a. The theorem only works for right-angled triangles, and c must be the hypotenuse.
Finding the hypotenuse. If you know both shorter sides, square them, add the squares, then take the square root. The square root undoes the squaring, because we want c, not c2.
Finding a shorter side. If you know the hypotenuse and one shorter side, square them, subtract (the hypotenuse square is the big one, so you take the smaller square away), then take the square root.
Why add one time and subtract the other? Because the equation is a2 + b2 = c2. Finding c means working out the right-hand side directly. Finding a means moving b2 to the other side, which turns the plus into a minus.
A calculator gives the square root of a number that is not a perfect square as a long decimal, so we round. Always read the question to see how accurate to be. The Higher tier also needs answers left as a surd (a square root like √40 simplified to 2√10) when the question asks for an exact answer.
Key ideas
- Hypotenuse = the side opposite the right angle = the longest side.
- Pythagoras: a2 + b2 = c2 (c is the hypotenuse).
- Finding the hypotenuse: add the squares, then square root.
- Finding a shorter side: subtract the squares, then square root.
- Check: the hypotenuse must be longer than both shorter sides.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. To find the hypotenuse, square the two shorter sides, add them, then take the square root.
- Spot the hypotenuse. The hypotenuse is the side opposite the right angle (the little square) and it is the longest side. The side we want, x, is the one opposite the right angle, so it is the hypotenuse. We know the two shorter sides.Shorter sides: 5 cm and 12 cm.
- Write down Pythagoras’ theorem and put the numbers in. For a right-angled triangle, (shorter side)2 + (shorter side)2 = (hypotenuse)2.a2 + b2 = c2
52 + 122 = x2 - Square each short side. Squaring means multiplying the number by itself.52 = 5 × 5 = 25
122 = 12 × 12 = 144 - Add the two squares. This gives x2, the square of the hypotenuse.x2 = 25 + 144 = 169
- Take the square root. We want x, not x2. The square root undoes the squaring.x = √169 = 13 cm
Check. The hypotenuse must be the longest side: 13 is bigger than 5 and 12. Also 52 + 122 = 169 = 132. ✓
Watch out. For the hypotenuse you add the squares. Do not stop at x2: the last step is the square root.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. To find the hypotenuse, square the two shorter sides, add them, then take the square root.
- Spot the hypotenuse. The hypotenuse is the side opposite the right angle (the little square) and it is the longest side. The side we want, x, is the one opposite the right angle, so it is the hypotenuse. We know the two shorter sides.Shorter sides: 7 cm and 9 cm.
- Write down Pythagoras’ theorem and put the numbers in. For a right-angled triangle, (shorter side)2 + (shorter side)2 = (hypotenuse)2.a2 + b2 = c2
72 + 92 = x2 - Square each short side. Squaring means multiplying the number by itself.72 = 7 × 7 = 49
92 = 9 × 9 = 81 - Add the two squares. This gives x2, the square of the hypotenuse.x2 = 49 + 81 = 130
- Take the square root. We want x, not x2. The square root undoes the squaring. Use the √ button on the calculator.x = √130 = 11.40175……
- Round to 1 d.p.. The question tells us how accurate to be.x = 11.4 cm
Check. Square the answer: 11.42 = 129.96, which is close to 130. The hypotenuse 11.4 is longer than both 7 and 9. ✓
Watch out. For the hypotenuse you add the squares. Do not stop at x2: the last step is the square root.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. To find a shorter side, square the hypotenuse and the known side, subtract, then take the square root.
- Spot the hypotenuse. The hypotenuse is opposite the right angle and is the longest side. This time it is given, and we are looking for one of the two shorter sides.Hypotenuse = 17 cm. The other known side = 8 cm. The side we want is x.
- Write down Pythagoras’ theorem and put the numbers in. The hypotenuse is always the one on its own on the right-hand side: (shorter)2 + (shorter)2 = (hypotenuse)2.a2 + b2 = c2
x2 + 82 = 172 - Square the known numbers. Squaring means multiplying the number by itself.82 = 8 × 8 = 64
172 = 17 × 17 = 289
x2 + 64 = 289 - Subtract 64 from both sides. We want x2 on its own. Taking 64 away from both sides keeps the equation balanced.x2 = 289 − 64 = 225
- Take the square root. We want x, not x2.x = √225 = 15 cm
Check. Put it back: 152 + 82 = 225 + 64 = 289 = 172. ✓ And 15 is shorter than the hypotenuse 17.
Watch out. When you are finding a shorter side you subtract. A shorter side must always be smaller than the hypotenuse; if your answer is bigger, you added by mistake.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. To find a shorter side, square the hypotenuse and the known side, subtract, then take the square root.
- Spot the hypotenuse. The hypotenuse is opposite the right angle and is the longest side. This time it is given, and we are looking for one of the two shorter sides.Hypotenuse = 9.4 m. The other known side = 6.2 m. The side we want is x.
- Write down Pythagoras’ theorem and put the numbers in. The hypotenuse is always the one on its own on the right-hand side: (shorter)2 + (shorter)2 = (hypotenuse)2.a2 + b2 = c2
x2 + 6.22 = 9.42 - Square the known numbers. Squaring means multiplying the number by itself.6.22 = 6.2 × 6.2 = 38.44
9.42 = 9.4 × 9.4 = 88.36
x2 + 38.44 = 88.36 - Subtract 38.44 from both sides. We want x2 on its own. Taking 38.44 away from both sides keeps the equation balanced.x2 = 88.36 − 38.44 = 49.92
- Take the square root. We want x, not x2.x = √49.92 = 7.06540…
- Round to 1 d.p.. The question tells us how accurate to be.x = 7.1 m
Check. Put it back: 7.12 + 6.22 = 50.41 + 38.44 = 88.85, which is close to 9.42 = 88.36. ✓ The answer is shorter than the hypotenuse 9.4.
Watch out. When you are finding a shorter side you subtract. A shorter side must always be smaller than the hypotenuse; if your answer is bigger, you added by mistake.
Watch out
Do not add the two shorter sides and stop, and do not forget the last step. If c2 = 169 the answer is c = 13, not 169. Also, always check that the answer for a shorter side is smaller than the hypotenuse. If it is bigger you added when you should have subtracted.Exercise 20A Pythagoras with whole-number answers
Non-calculatorMethod. Squaring a number means multiplying it by itself.
- Write the multiplication. 42 means 4 × 4.42 = 4 × 4
- Work it out. 4 × 4 = 16
Check. √16 = 4, so squaring and square rooting undo each other. ✓
Watch out. 42 is 4 × 4, not 4 × 2.
Method. Squaring a number means multiplying it by itself.
- Write the multiplication. 72 means 7 × 7.72 = 7 × 7
- Work it out. 7 × 7 = 49
Check. √49 = 7, so squaring and square rooting undo each other. ✓
Watch out. 72 is 7 × 7, not 7 × 2.
Method. Squaring a number means multiplying it by itself.
- Write the multiplication. 92 means 9 × 9.92 = 9 × 9
- Work it out. 9 × 9 = 81
Check. √81 = 9, so squaring and square rooting undo each other. ✓
Watch out. 92 is 9 × 9, not 9 × 2.
Method. Squaring a number means multiplying it by itself.
- Write the multiplication. 122 means 12 × 12.122 = 12 × 12
- Work it out. 12 × 12 = 144
Check. √144 = 12, so squaring and square rooting undo each other. ✓
Watch out. 122 is 12 × 12, not 12 × 2.
Method. Squaring a number means multiplying it by itself.
- Write the multiplication. 152 means 15 × 15.152 = 15 × 15
- Work it out. 15 × 15 = 225
Check. √225 = 15, so squaring and square rooting undo each other. ✓
Watch out. 152 is 15 × 15, not 15 × 2.
Method. Squaring a number means multiplying it by itself.
- Write the multiplication. 202 means 20 × 20.202 = 20 × 20
- Work it out. 20 × 20 = 400
Check. √400 = 20, so squaring and square rooting undo each other. ✓
Watch out. 202 is 20 × 20, not 20 × 2.
Method. The square root asks: which number multiplied by itself gives this?
- Think of square numbers. We want a number that gives 25 when multiplied by itself.5 × 5 = 25
- Write the answer. √25 = 5
Check. 52 = 25. ✓
Watch out. The square root of 36 is 6, not 18 (that is half of 36).
Method. The square root asks: which number multiplied by itself gives this?
- Think of square numbers. We want a number that gives 49 when multiplied by itself.7 × 7 = 49
- Write the answer. √49 = 7
Check. 72 = 49. ✓
Watch out. The square root of 36 is 6, not 18 (that is half of 36).
Method. The square root asks: which number multiplied by itself gives this?
- Think of square numbers. We want a number that gives 81 when multiplied by itself.9 × 9 = 81
- Write the answer. √81 = 9
Check. 92 = 81. ✓
Watch out. The square root of 36 is 6, not 18 (that is half of 36).
Method. The square root asks: which number multiplied by itself gives this?
- Think of square numbers. We want a number that gives 144 when multiplied by itself.12 × 12 = 144
- Write the answer. √144 = 12
Check. 122 = 144. ✓
Watch out. The square root of 36 is 6, not 18 (that is half of 36).
Method. The square root asks: which number multiplied by itself gives this?
- Think of square numbers. We want a number that gives 169 when multiplied by itself.13 × 13 = 169
- Write the answer. √169 = 13
Check. 132 = 169. ✓
Watch out. The square root of 36 is 6, not 18 (that is half of 36).
Method. The square root asks: which number multiplied by itself gives this?
- Think of square numbers. We want a number that gives 225 when multiplied by itself.15 × 15 = 225
- Write the answer. √225 = 15
Check. 152 = 225. ✓
Watch out. The square root of 36 is 6, not 18 (that is half of 36).
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. The hypotenuse is the side opposite the right angle. It does not touch the right angle.
- Find the right angle. The little square marks the 90° angle.The right angle is at P.
- Find the side opposite it. The side opposite a vertex is the one that does not have that letter. The two sides that touch P are PQ and PR.The side that does not touch P is QR.
- Check it is the longest side. The hypotenuse is always the longest side of a right-angled triangle.QR is the longest side in the picture. The hypotenuse is QR.
Check. The longest side of a right-angled triangle is always opposite the right angle. ✓
Watch out. The hypotenuse is not decided by which way up the triangle is drawn. Always find the right angle first.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. The hypotenuse is the side opposite the right angle. It does not touch the right angle.
- Find the right angle. The little square marks the 90° angle.The right angle is at A.
- Find the side opposite it. The side opposite a vertex is the one that does not have that letter. The two sides that touch A are AB and AC.The side that does not touch A is BC.
- Check it is the longest side. The hypotenuse is always the longest side of a right-angled triangle.BC is the longest side in the picture. The hypotenuse is BC.
Check. The longest side of a right-angled triangle is always opposite the right angle. ✓
Watch out. The hypotenuse is not decided by which way up the triangle is drawn. Always find the right angle first.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. The hypotenuse is the side opposite the right angle. It does not touch the right angle.
- Find the right angle. The little square marks the 90° angle.The right angle is at X.
- Find the side opposite it. The side opposite a vertex is the one that does not have that letter. The two sides that touch X are XY and XZ.The side that does not touch X is YZ.
- Check it is the longest side. The hypotenuse is always the longest side of a right-angled triangle.YZ is the longest side in the picture. The hypotenuse is YZ.
Check. The longest side of a right-angled triangle is always opposite the right angle. ✓
Watch out. The hypotenuse is not decided by which way up the triangle is drawn. Always find the right angle first.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. To find the hypotenuse, square the two shorter sides, add them, then take the square root.
- Spot the hypotenuse. The hypotenuse is the side opposite the right angle (the little square) and it is the longest side. The side we want, x, is the one opposite the right angle, so it is the hypotenuse. We know the two shorter sides.Shorter sides: 3 cm and 4 cm.
- Write down Pythagoras’ theorem and put the numbers in. For a right-angled triangle, (shorter side)2 + (shorter side)2 = (hypotenuse)2.a2 + b2 = c2
32 + 42 = x2 - Square each short side. Squaring means multiplying the number by itself.32 = 3 × 3 = 9
42 = 4 × 4 = 16 - Add the two squares. This gives x2, the square of the hypotenuse.x2 = 9 + 16 = 25
- Take the square root. We want x, not x2. The square root undoes the squaring.x = √25 = 5 cm
Check. The hypotenuse must be the longest side: 5 is bigger than 3 and 4. Also 32 + 42 = 25 = 52. ✓
Watch out. For the hypotenuse you add the squares. Do not stop at x2: the last step is the square root.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. To find the hypotenuse, square the two shorter sides, add them, then take the square root.
- Spot the hypotenuse. The hypotenuse is the side opposite the right angle (the little square) and it is the longest side. The side we want, x, is the one opposite the right angle, so it is the hypotenuse. We know the two shorter sides.Shorter sides: 6 cm and 8 cm.
- Write down Pythagoras’ theorem and put the numbers in. For a right-angled triangle, (shorter side)2 + (shorter side)2 = (hypotenuse)2.a2 + b2 = c2
62 + 82 = x2 - Square each short side. Squaring means multiplying the number by itself.62 = 6 × 6 = 36
82 = 8 × 8 = 64 - Add the two squares. This gives x2, the square of the hypotenuse.x2 = 36 + 64 = 100
- Take the square root. We want x, not x2. The square root undoes the squaring.x = √100 = 10 cm
Check. The hypotenuse must be the longest side: 10 is bigger than 6 and 8. Also 62 + 82 = 100 = 102. ✓
Watch out. For the hypotenuse you add the squares. Do not stop at x2: the last step is the square root.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. To find the hypotenuse, square the two shorter sides, add them, then take the square root.
- Spot the hypotenuse. The hypotenuse is the side opposite the right angle (the little square) and it is the longest side. The side we want, x, is the one opposite the right angle, so it is the hypotenuse. We know the two shorter sides.Shorter sides: 5 cm and 12 cm.
- Write down Pythagoras’ theorem and put the numbers in. For a right-angled triangle, (shorter side)2 + (shorter side)2 = (hypotenuse)2.a2 + b2 = c2
52 + 122 = x2 - Square each short side. Squaring means multiplying the number by itself.52 = 5 × 5 = 25
122 = 12 × 12 = 144 - Add the two squares. This gives x2, the square of the hypotenuse.x2 = 25 + 144 = 169
- Take the square root. We want x, not x2. The square root undoes the squaring.x = √169 = 13 cm
Check. The hypotenuse must be the longest side: 13 is bigger than 5 and 12. Also 52 + 122 = 169 = 132. ✓
Watch out. For the hypotenuse you add the squares. Do not stop at x2: the last step is the square root.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. To find the hypotenuse, square the two shorter sides, add them, then take the square root.
- Spot the hypotenuse. The hypotenuse is the side opposite the right angle (the little square) and it is the longest side. The side we want, x, is the one opposite the right angle, so it is the hypotenuse. We know the two shorter sides.Shorter sides: 8 cm and 15 cm.
- Write down Pythagoras’ theorem and put the numbers in. For a right-angled triangle, (shorter side)2 + (shorter side)2 = (hypotenuse)2.a2 + b2 = c2
82 + 152 = x2 - Square each short side. Squaring means multiplying the number by itself.82 = 8 × 8 = 64
152 = 15 × 15 = 225 - Add the two squares. This gives x2, the square of the hypotenuse.x2 = 64 + 225 = 289
- Take the square root. We want x, not x2. The square root undoes the squaring.x = √289 = 17 cm
Check. The hypotenuse must be the longest side: 17 is bigger than 8 and 15. Also 82 + 152 = 289 = 172. ✓
Watch out. For the hypotenuse you add the squares. Do not stop at x2: the last step is the square root.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. To find the hypotenuse, square the two shorter sides, add them, then take the square root.
- Spot the hypotenuse. The hypotenuse is the side opposite the right angle (the little square) and it is the longest side. The side we want, x, is the one opposite the right angle, so it is the hypotenuse. We know the two shorter sides.Shorter sides: 20 cm and 21 cm.
- Write down Pythagoras’ theorem and put the numbers in. For a right-angled triangle, (shorter side)2 + (shorter side)2 = (hypotenuse)2.a2 + b2 = c2
202 + 212 = x2 - Square each short side. Squaring means multiplying the number by itself.202 = 20 × 20 = 400
212 = 21 × 21 = 441 - Add the two squares. This gives x2, the square of the hypotenuse.x2 = 400 + 441 = 841
- Take the square root. We want x, not x2. The square root undoes the squaring.x = √841 = 29 cm
Check. The hypotenuse must be the longest side: 29 is bigger than 20 and 21. Also 202 + 212 = 841 = 292. ✓
Watch out. For the hypotenuse you add the squares. Do not stop at x2: the last step is the square root.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. To find the hypotenuse, square the two shorter sides, add them, then take the square root.
- Spot the hypotenuse. The hypotenuse is the side opposite the right angle (the little square) and it is the longest side. The side we want, x, is the one opposite the right angle, so it is the hypotenuse. We know the two shorter sides.Shorter sides: 24 cm and 7 cm.
- Write down Pythagoras’ theorem and put the numbers in. For a right-angled triangle, (shorter side)2 + (shorter side)2 = (hypotenuse)2.a2 + b2 = c2
242 + 72 = x2 - Square each short side. Squaring means multiplying the number by itself.242 = 24 × 24 = 576
72 = 7 × 7 = 49 - Add the two squares. This gives x2, the square of the hypotenuse.x2 = 576 + 49 = 625
- Take the square root. We want x, not x2. The square root undoes the squaring.x = √625 = 25 cm
Check. The hypotenuse must be the longest side: 25 is bigger than 24 and 7. Also 242 + 72 = 625 = 252. ✓
Watch out. For the hypotenuse you add the squares. Do not stop at x2: the last step is the square root.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. To find a shorter side, square the hypotenuse and the known side, subtract, then take the square root.
- Spot the hypotenuse. The hypotenuse is opposite the right angle and is the longest side. This time it is given, and we are looking for one of the two shorter sides.Hypotenuse = 13 cm. The other known side = 5 cm. The side we want is x.
- Write down Pythagoras’ theorem and put the numbers in. The hypotenuse is always the one on its own on the right-hand side: (shorter)2 + (shorter)2 = (hypotenuse)2.a2 + b2 = c2
x2 + 52 = 132 - Square the known numbers. Squaring means multiplying the number by itself.52 = 5 × 5 = 25
132 = 13 × 13 = 169
x2 + 25 = 169 - Subtract 25 from both sides. We want x2 on its own. Taking 25 away from both sides keeps the equation balanced.x2 = 169 − 25 = 144
- Take the square root. We want x, not x2.x = √144 = 12 cm
Check. Put it back: 122 + 52 = 144 + 25 = 169 = 132. ✓ And 12 is shorter than the hypotenuse 13.
Watch out. When you are finding a shorter side you subtract. A shorter side must always be smaller than the hypotenuse; if your answer is bigger, you added by mistake.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. To find a shorter side, square the hypotenuse and the known side, subtract, then take the square root.
- Spot the hypotenuse. The hypotenuse is opposite the right angle and is the longest side. This time it is given, and we are looking for one of the two shorter sides.Hypotenuse = 10 cm. The other known side = 6 cm. The side we want is x.
- Write down Pythagoras’ theorem and put the numbers in. The hypotenuse is always the one on its own on the right-hand side: (shorter)2 + (shorter)2 = (hypotenuse)2.a2 + b2 = c2
x2 + 62 = 102 - Square the known numbers. Squaring means multiplying the number by itself.62 = 6 × 6 = 36
102 = 10 × 10 = 100
x2 + 36 = 100 - Subtract 36 from both sides. We want x2 on its own. Taking 36 away from both sides keeps the equation balanced.x2 = 100 − 36 = 64
- Take the square root. We want x, not x2.x = √64 = 8 cm
Check. Put it back: 82 + 62 = 64 + 36 = 100 = 102. ✓ And 8 is shorter than the hypotenuse 10.
Watch out. When you are finding a shorter side you subtract. A shorter side must always be smaller than the hypotenuse; if your answer is bigger, you added by mistake.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. To find a shorter side, square the hypotenuse and the known side, subtract, then take the square root.
- Spot the hypotenuse. The hypotenuse is opposite the right angle and is the longest side. This time it is given, and we are looking for one of the two shorter sides.Hypotenuse = 25 cm. The other known side = 24 cm. The side we want is x.
- Write down Pythagoras’ theorem and put the numbers in. The hypotenuse is always the one on its own on the right-hand side: (shorter)2 + (shorter)2 = (hypotenuse)2.a2 + b2 = c2
x2 + 242 = 252 - Square the known numbers. Squaring means multiplying the number by itself.242 = 24 × 24 = 576
252 = 25 × 25 = 625
x2 + 576 = 625 - Subtract 576 from both sides. We want x2 on its own. Taking 576 away from both sides keeps the equation balanced.x2 = 625 − 576 = 49
- Take the square root. We want x, not x2.x = √49 = 7 cm
Check. Put it back: 72 + 242 = 49 + 576 = 625 = 252. ✓ And 7 is shorter than the hypotenuse 25.
Watch out. When you are finding a shorter side you subtract. A shorter side must always be smaller than the hypotenuse; if your answer is bigger, you added by mistake.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. To find a shorter side, square the hypotenuse and the known side, subtract, then take the square root.
- Spot the hypotenuse. The hypotenuse is opposite the right angle and is the longest side. This time it is given, and we are looking for one of the two shorter sides.Hypotenuse = 17 cm. The other known side = 15 cm. The side we want is x.
- Write down Pythagoras’ theorem and put the numbers in. The hypotenuse is always the one on its own on the right-hand side: (shorter)2 + (shorter)2 = (hypotenuse)2.a2 + b2 = c2
x2 + 152 = 172 - Square the known numbers. Squaring means multiplying the number by itself.152 = 15 × 15 = 225
172 = 17 × 17 = 289
x2 + 225 = 289 - Subtract 225 from both sides. We want x2 on its own. Taking 225 away from both sides keeps the equation balanced.x2 = 289 − 225 = 64
- Take the square root. We want x, not x2.x = √64 = 8 cm
Check. Put it back: 82 + 152 = 64 + 225 = 289 = 172. ✓ And 8 is shorter than the hypotenuse 17.
Watch out. When you are finding a shorter side you subtract. A shorter side must always be smaller than the hypotenuse; if your answer is bigger, you added by mistake.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. To find a shorter side, square the hypotenuse and the known side, subtract, then take the square root.
- Spot the hypotenuse. The hypotenuse is opposite the right angle and is the longest side. This time it is given, and we are looking for one of the two shorter sides.Hypotenuse = 41 cm. The other known side = 40 cm. The side we want is x.
- Write down Pythagoras’ theorem and put the numbers in. The hypotenuse is always the one on its own on the right-hand side: (shorter)2 + (shorter)2 = (hypotenuse)2.a2 + b2 = c2
x2 + 402 = 412 - Square the known numbers. Squaring means multiplying the number by itself.402 = 40 × 40 = 1600
412 = 41 × 41 = 1681
x2 + 1600 = 1681 - Subtract 1600 from both sides. We want x2 on its own. Taking 1600 away from both sides keeps the equation balanced.x2 = 1681 − 1600 = 81
- Take the square root. We want x, not x2.x = √81 = 9 cm
Check. Put it back: 92 + 402 = 81 + 1600 = 1681 = 412. ✓ And 9 is shorter than the hypotenuse 41.
Watch out. When you are finding a shorter side you subtract. A shorter side must always be smaller than the hypotenuse; if your answer is bigger, you added by mistake.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. To find a shorter side, square the hypotenuse and the known side, subtract, then take the square root.
- Draw a right-angled triangle. The ladder, the wall and the ground make a right-angled triangle. The ladder is the hypotenuse because it is opposite the right angle between wall and ground.Ladder = 13 m (hypotenuse). Foot of the ladder to the wall = 5 m. Height up the wall = h.
- Spot the hypotenuse. The hypotenuse is opposite the right angle and is the longest side. This time it is given, and we are looking for one of the two shorter sides.Hypotenuse = 13 m. The other known side = 5 m. The side we want is h.
- Write down Pythagoras’ theorem and put the numbers in. The hypotenuse is always the one on its own on the right-hand side: (shorter)2 + (shorter)2 = (hypotenuse)2.a2 + b2 = c2
h2 + 52 = 132 - Square the known numbers. Squaring means multiplying the number by itself.52 = 5 × 5 = 25
132 = 13 × 13 = 169
h2 + 25 = 169 - Subtract 25 from both sides. We want h2 on its own. Taking 25 away from both sides keeps the equation balanced.h2 = 169 − 25 = 144
- Take the square root. We want h, not h2.h = √144 = 12 m
Check. Put it back: 122 + 52 = 144 + 25 = 169 = 132. ✓ And 12 is shorter than the hypotenuse 13.
Watch out. When you are finding a shorter side you subtract. A shorter side must always be smaller than the hypotenuse; if your answer is bigger, you added by mistake.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. To find the hypotenuse, square the two shorter sides, add them, then take the square root.
- Find a right-angled triangle. The diagonal of a rectangle cuts it into two right-angled triangles, because every corner of a rectangle is 90°. The diagonal is the hypotenuse.Shorter sides: 24 cm and 10 cm.
- Spot the hypotenuse. The hypotenuse is the side opposite the right angle (the little square) and it is the longest side. The side we want, x, is the one opposite the right angle, so it is the hypotenuse. We know the two shorter sides.Shorter sides: 24 cm and 10 cm.
- Write down Pythagoras’ theorem and put the numbers in. For a right-angled triangle, (shorter side)2 + (shorter side)2 = (hypotenuse)2.a2 + b2 = c2
242 + 102 = x2 - Square each short side. Squaring means multiplying the number by itself.242 = 24 × 24 = 576
102 = 10 × 10 = 100 - Add the two squares. This gives x2, the square of the hypotenuse.x2 = 576 + 100 = 676
- Take the square root. We want x, not x2. The square root undoes the squaring.x = √676 = 26 cm
Check. The hypotenuse must be the longest side: 26 is bigger than 24 and 10. Also 242 + 102 = 676 = 262. ✓
Watch out. For the hypotenuse you add the squares. Do not stop at x2: the last step is the square root.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. To find the hypotenuse, square the two shorter sides, add them, then take the square root.
- Spot the hypotenuse. The hypotenuse is the side opposite the right angle (the little square) and it is the longest side. The side we want, x, is the one opposite the right angle, so it is the hypotenuse. We know the two shorter sides.Shorter sides: 2 cm and 3 cm.
- Write down Pythagoras’ theorem and put the numbers in. For a right-angled triangle, (shorter side)2 + (shorter side)2 = (hypotenuse)2.a2 + b2 = c2
22 + 32 = x2 - Square each short side. Squaring means multiplying the number by itself.22 = 2 × 2 = 4
32 = 3 × 3 = 9 - Add the two squares. This gives x2, the square of the hypotenuse.x2 = 4 + 9 = 13
- Take the square root and leave it exact. The question wants an exact answer, so we do not use a decimal. We simplify the surd instead.x = √13
√13 cannot be simplified (no square factor), so it stays as √13.
x = √13 cm
Check. On a calculator √13 = 3.60555… and √13 = 3.60555…. It is longer than 2 and 3. ✓
Watch out. For the hypotenuse you add the squares. Do not stop at x2: the last step is the square root.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. To find the hypotenuse, square the two shorter sides, add them, then take the square root.
- Spot the hypotenuse. The hypotenuse is the side opposite the right angle (the little square) and it is the longest side. The side we want, x, is the one opposite the right angle, so it is the hypotenuse. We know the two shorter sides.Shorter sides: 1 cm and 5 cm.
- Write down Pythagoras’ theorem and put the numbers in. For a right-angled triangle, (shorter side)2 + (shorter side)2 = (hypotenuse)2.a2 + b2 = c2
12 + 52 = x2 - Square each short side. Squaring means multiplying the number by itself.12 = 1 × 1 = 1
52 = 5 × 5 = 25 - Add the two squares. This gives x2, the square of the hypotenuse.x2 = 1 + 25 = 26
- Take the square root and leave it exact. The question wants an exact answer, so we do not use a decimal. We simplify the surd instead.x = √26
√26 cannot be simplified (no square factor), so it stays as √26.
x = √26 cm
Check. On a calculator √26 = 5.09901… and √26 = 5.09901…. It is longer than 1 and 5. ✓
Watch out. For the hypotenuse you add the squares. Do not stop at x2: the last step is the square root.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. Aisha added the lengths. Pythagoras’ theorem adds the squares of the lengths, then takes a square root.
- Say what is wrong. The theorem is a2 + b2 = c2. It does not say a + b = c. In any triangle the longest side is shorter than the other two sides added together.Aisha added 6 + 8 = 14, which is not allowed.
- Do the calculation properly. Square both, add, then square root.62 + 82 = 36 + 64 = 100
√100 = 10 - Write the answer. The hypotenuse is 10 cm, which is less than 14 cm.
Check. 6, 8, 10 is double the triangle 3, 4, 5, so 10 is right. ✓
Watch out. Squares first, then add, then square root.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. The hypotenuse is the longest side of a right-angled triangle, so it must be longer than each shorter side.
- Compare with the shorter sides. A shorter side is 10 cm. The hypotenuse has to be longer than 10 cm, but Kim says 9 cm.9 cm is less than 10 cm, so Kim is wrong.
- Find the correct length. Square, add, then take the square root.102 + 62 = 100 + 36 = 136
√136 = 11.66190… - Round. To 1 decimal place.Hypotenuse = 11.7 cm
Check. 11.7 is longer than 10 and longer than 6. ✓
Watch out. Always ask: is my hypotenuse the longest side? If not, there is a mistake.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. The perimeter is all three sides added. We know two of them, so find the third with Pythagoras’ theorem.
- Find the missing shorter side. The hypotenuse (25 cm) is given, so we subtract the squares.x2 + 72 = 252
x2 = 625 − 49 = 576
x = √576 = 24 cm - Add the three sides. The perimeter is the distance all the way round.7 + 24 + 25 = 56 cm
- State the conclusion. A “show that” question needs a final statement.The perimeter is 56 cm, as required.
Check. 7, 24, 25 is a Pythagorean triple: 49 + 576 = 625. ✓
Watch out. In a “show that” question, show every line of working and finish with a conclusion.
Exercise 20B Pythagoras with a calculator
Calculator allowedKey words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. To find the hypotenuse, square the two shorter sides, add them, then take the square root.
- Spot the hypotenuse. The hypotenuse is the side opposite the right angle (the little square) and it is the longest side. The side we want, x, is the one opposite the right angle, so it is the hypotenuse. We know the two shorter sides.Shorter sides: 4 cm and 7 cm.
- Write down Pythagoras’ theorem and put the numbers in. For a right-angled triangle, (shorter side)2 + (shorter side)2 = (hypotenuse)2.a2 + b2 = c2
42 + 72 = x2 - Square each short side. Squaring means multiplying the number by itself.42 = 4 × 4 = 16
72 = 7 × 7 = 49 - Add the two squares. This gives x2, the square of the hypotenuse.x2 = 16 + 49 = 65
- Take the square root. We want x, not x2. The square root undoes the squaring. Use the √ button on the calculator.x = √65 = 8.06225……
- Round to 1 d.p.. The question tells us how accurate to be.x = 8.1 cm
Check. Square the answer: 8.12 = 65.61, which is close to 65. The hypotenuse 8.1 is longer than both 4 and 7. ✓
Watch out. For the hypotenuse you add the squares. Do not stop at x2: the last step is the square root.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. To find the hypotenuse, square the two shorter sides, add them, then take the square root.
- Spot the hypotenuse. The hypotenuse is the side opposite the right angle (the little square) and it is the longest side. The side we want, x, is the one opposite the right angle, so it is the hypotenuse. We know the two shorter sides.Shorter sides: 6 cm and 9 cm.
- Write down Pythagoras’ theorem and put the numbers in. For a right-angled triangle, (shorter side)2 + (shorter side)2 = (hypotenuse)2.a2 + b2 = c2
62 + 92 = x2 - Square each short side. Squaring means multiplying the number by itself.62 = 6 × 6 = 36
92 = 9 × 9 = 81 - Add the two squares. This gives x2, the square of the hypotenuse.x2 = 36 + 81 = 117
- Take the square root. We want x, not x2. The square root undoes the squaring. Use the √ button on the calculator.x = √117 = 10.81665……
- Round to 1 d.p.. The question tells us how accurate to be.x = 10.8 cm
Check. Square the answer: 10.82 = 116.64, which is close to 117. The hypotenuse 10.8 is longer than both 6 and 9. ✓
Watch out. For the hypotenuse you add the squares. Do not stop at x2: the last step is the square root.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. To find the hypotenuse, square the two shorter sides, add them, then take the square root.
- Spot the hypotenuse. The hypotenuse is the side opposite the right angle (the little square) and it is the longest side. The side we want, x, is the one opposite the right angle, so it is the hypotenuse. We know the two shorter sides.Shorter sides: 8.5 cm and 5.2 cm.
- Write down Pythagoras’ theorem and put the numbers in. For a right-angled triangle, (shorter side)2 + (shorter side)2 = (hypotenuse)2.a2 + b2 = c2
8.52 + 5.22 = x2 - Square each short side. Squaring means multiplying the number by itself.8.52 = 8.5 × 8.5 = 72.25
5.22 = 5.2 × 5.2 = 27.04 - Add the two squares. This gives x2, the square of the hypotenuse.x2 = 72.25 + 27.04 = 99.29
- Take the square root. We want x, not x2. The square root undoes the squaring. Use the √ button on the calculator.x = √99.29 = 9.96443……
- Round to 1 d.p.. The question tells us how accurate to be.x = 10.0 cm
Check. Square the answer: 10.02 = 100.00, which is close to 99.29. The hypotenuse 10.0 is longer than both 8.5 and 5.2. ✓
Watch out. For the hypotenuse you add the squares. Do not stop at x2: the last step is the square root.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. To find the hypotenuse, square the two shorter sides, add them, then take the square root.
- Spot the hypotenuse. The hypotenuse is the side opposite the right angle (the little square) and it is the longest side. The side we want, x, is the one opposite the right angle, so it is the hypotenuse. We know the two shorter sides.Shorter sides: 12 cm and 12 cm.
- Write down Pythagoras’ theorem and put the numbers in. For a right-angled triangle, (shorter side)2 + (shorter side)2 = (hypotenuse)2.a2 + b2 = c2
122 + 122 = x2 - Square each short side. Squaring means multiplying the number by itself.122 = 12 × 12 = 144
122 = 12 × 12 = 144 - Add the two squares. This gives x2, the square of the hypotenuse.x2 = 144 + 144 = 288
- Take the square root. We want x, not x2. The square root undoes the squaring. Use the √ button on the calculator.x = √288 = 16.97056……
- Round to 1 d.p.. The question tells us how accurate to be.x = 17.0 cm
Check. Square the answer: 17.02 = 289.00, which is close to 288. The hypotenuse 17.0 is longer than both 12 and 12. ✓
Watch out. For the hypotenuse you add the squares. Do not stop at x2: the last step is the square root.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. To find a shorter side, square the hypotenuse and the known side, subtract, then take the square root.
- Spot the hypotenuse. The hypotenuse is opposite the right angle and is the longest side. This time it is given, and we are looking for one of the two shorter sides.Hypotenuse = 9 cm. The other known side = 5 cm. The side we want is x.
- Write down Pythagoras’ theorem and put the numbers in. The hypotenuse is always the one on its own on the right-hand side: (shorter)2 + (shorter)2 = (hypotenuse)2.a2 + b2 = c2
x2 + 52 = 92 - Square the known numbers. Squaring means multiplying the number by itself.52 = 5 × 5 = 25
92 = 9 × 9 = 81
x2 + 25 = 81 - Subtract 25 from both sides. We want x2 on its own. Taking 25 away from both sides keeps the equation balanced.x2 = 81 − 25 = 56
- Take the square root. We want x, not x2.x = √56 = 7.48331…
- Round to 1 d.p.. The question tells us how accurate to be.x = 7.5 cm
Check. Put it back: 7.52 + 52 = 56.25 + 25 = 81.25, which is close to 92 = 81. ✓ The answer is shorter than the hypotenuse 9.
Watch out. When you are finding a shorter side you subtract. A shorter side must always be smaller than the hypotenuse; if your answer is bigger, you added by mistake.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. To find a shorter side, square the hypotenuse and the known side, subtract, then take the square root.
- Spot the hypotenuse. The hypotenuse is opposite the right angle and is the longest side. This time it is given, and we are looking for one of the two shorter sides.Hypotenuse = 14 cm. The other known side = 8.4 cm. The side we want is x.
- Write down Pythagoras’ theorem and put the numbers in. The hypotenuse is always the one on its own on the right-hand side: (shorter)2 + (shorter)2 = (hypotenuse)2.a2 + b2 = c2
x2 + 8.42 = 142 - Square the known numbers. Squaring means multiplying the number by itself.8.42 = 8.4 × 8.4 = 70.56
142 = 14 × 14 = 196
x2 + 70.56 = 196 - Subtract 70.56 from both sides. We want x2 on its own. Taking 70.56 away from both sides keeps the equation balanced.x2 = 196 − 70.56 = 125.44
- Take the square root. We want x, not x2.x = √125.44 = 11.2 cm
Check. Put it back: 11.22 + 8.42 = 125.44 + 70.56 = 196 = 142. ✓ And 11.2 is shorter than the hypotenuse 14.
Watch out. When you are finding a shorter side you subtract. A shorter side must always be smaller than the hypotenuse; if your answer is bigger, you added by mistake.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. To find a shorter side, square the hypotenuse and the known side, subtract, then take the square root.
- Spot the hypotenuse. The hypotenuse is opposite the right angle and is the longest side. This time it is given, and we are looking for one of the two shorter sides.Hypotenuse = 7.5 cm. The other known side = 6 cm. The side we want is x.
- Write down Pythagoras’ theorem and put the numbers in. The hypotenuse is always the one on its own on the right-hand side: (shorter)2 + (shorter)2 = (hypotenuse)2.a2 + b2 = c2
x2 + 62 = 7.52 - Square the known numbers. Squaring means multiplying the number by itself.62 = 6 × 6 = 36
7.52 = 7.5 × 7.5 = 56.25
x2 + 36 = 56.25 - Subtract 36 from both sides. We want x2 on its own. Taking 36 away from both sides keeps the equation balanced.x2 = 56.25 − 36 = 20.25
- Take the square root. We want x, not x2.x = √20.25 = 4.5 cm
Check. Put it back: 4.52 + 62 = 20.25 + 36 = 56.25 = 7.52. ✓ And 4.5 is shorter than the hypotenuse 7.5.
Watch out. When you are finding a shorter side you subtract. A shorter side must always be smaller than the hypotenuse; if your answer is bigger, you added by mistake.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. To find a shorter side, square the hypotenuse and the known side, subtract, then take the square root.
- Spot the hypotenuse. The hypotenuse is opposite the right angle and is the longest side. This time it is given, and we are looking for one of the two shorter sides.Hypotenuse = 15 cm. The other known side = 4.2 cm. The side we want is x.
- Write down Pythagoras’ theorem and put the numbers in. The hypotenuse is always the one on its own on the right-hand side: (shorter)2 + (shorter)2 = (hypotenuse)2.a2 + b2 = c2
x2 + 4.22 = 152 - Square the known numbers. Squaring means multiplying the number by itself.4.22 = 4.2 × 4.2 = 17.64
152 = 15 × 15 = 225
x2 + 17.64 = 225 - Subtract 17.64 from both sides. We want x2 on its own. Taking 17.64 away from both sides keeps the equation balanced.x2 = 225 − 17.64 = 207.36
- Take the square root. We want x, not x2.x = √207.36 = 14.4 cm
Check. Put it back: 14.42 + 4.22 = 207.36 + 17.64 = 225 = 152. ✓ And 14.4 is shorter than the hypotenuse 15.
Watch out. When you are finding a shorter side you subtract. A shorter side must always be smaller than the hypotenuse; if your answer is bigger, you added by mistake.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. To find the hypotenuse, square the two shorter sides, add them, then take the square root.
- Spot the hypotenuse. The hypotenuse is the side opposite the right angle (the little square) and it is the longest side. The side we want, x, is the one opposite the right angle, so it is the hypotenuse. We know the two shorter sides.Shorter sides: 15.3 m and 8.7 m.
- Write down Pythagoras’ theorem and put the numbers in. For a right-angled triangle, (shorter side)2 + (shorter side)2 = (hypotenuse)2.a2 + b2 = c2
15.32 + 8.72 = x2 - Square each short side. Squaring means multiplying the number by itself.15.32 = 15.3 × 15.3 = 234.09
8.72 = 8.7 × 8.7 = 75.69 - Add the two squares. This gives x2, the square of the hypotenuse.x2 = 234.09 + 75.69 = 309.78
- Take the square root. We want x, not x2. The square root undoes the squaring. Use the √ button on the calculator.x = √309.78 = 17.60056……
- Round to 2 d.p.. The question tells us how accurate to be.x = 17.60 m
Check. Square the answer: 17.602 = 309.76, which is close to 309.78. The hypotenuse 17.60 is longer than both 15.3 and 8.7. ✓
Watch out. For the hypotenuse you add the squares. Do not stop at x2: the last step is the square root.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. To find a shorter side, square the hypotenuse and the known side, subtract, then take the square root.
- Spot the hypotenuse. The hypotenuse is opposite the right angle and is the longest side. This time it is given, and we are looking for one of the two shorter sides.Hypotenuse = 23.6 mm. The other known side = 19.4 mm. The side we want is x.
- Write down Pythagoras’ theorem and put the numbers in. The hypotenuse is always the one on its own on the right-hand side: (shorter)2 + (shorter)2 = (hypotenuse)2.a2 + b2 = c2
x2 + 19.42 = 23.62 - Square the known numbers. Squaring means multiplying the number by itself.19.42 = 19.4 × 19.4 = 376.36
23.62 = 23.6 × 23.6 = 556.96
x2 + 376.36 = 556.96 - Subtract 376.36 from both sides. We want x2 on its own. Taking 376.36 away from both sides keeps the equation balanced.x2 = 556.96 − 376.36 = 180.6
- Take the square root. We want x, not x2.x = √180.6 = 13.43874…
- Round to 3 s.f.. The question tells us how accurate to be.x = 13.4 mm
Check. Put it back: 13.42 + 19.42 = 179.56 + 376.36 = 555.92, which is close to 23.62 = 556.96. ✓ The answer is shorter than the hypotenuse 23.6.
Watch out. When you are finding a shorter side you subtract. A shorter side must always be smaller than the hypotenuse; if your answer is bigger, you added by mistake.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. Use Pythagoras’ theorem to find the diagonal, then convert from centimetres to inches.
- Draw the right-angled triangle. The diagonal is the hypotenuse of a right-angled triangle with shorter sides 80 cm and 45 cm.Shorter sides 80 cm and 45 cm.
- Square, add, square root. Hypotenuse: add the squares.802 + 452 = 6400 + 2025 = 8425
√8425 = 91.78779… - Round to the nearest cm. Nearest whole number.Diagonal = 92 cm
- Convert to inches. There are 2.54 cm in 1 inch, so divide by 2.54.92 ÷ 2.54 = 36.22047… ≈ 36 inches
Check. A 36 inch screen is the usual name for this size of screen. 36 × 2.54 = 91.4 cm, close to 92 cm. ✓
Watch out. Convert at the end. Do not round the diagonal before you need to.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. Find the length of the diagonal with Pythagoras’ theorem, then compare it with the distance along the two edges.
- Distance along the edges. He walks the length and then the width.120 + 90 = 210 m
- Distance along the diagonal. The diagonal is the hypotenuse of a right-angled triangle with shorter sides 120 m and 90 m.1202 + 902 = 14 400 + 8100 = 22 500
√22 500 = 150 - Find the difference. Take the shorter route from the longer.210 − 150 = 60 m
Check. 120 and 90 are 30 times 4 and 3, so the diagonal is 30 times 5 = 150. ✓
Watch out. The answer is the difference between the routes, not the length of the diagonal.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. To find the hypotenuse, square the two shorter sides, add them, then take the square root.
- Draw the path. North and east make a right angle, so the ship’s path and the straight line back to the start form a right-angled triangle.Shorter sides 15 km and 8 km. The distance d is the hypotenuse.
- Spot the hypotenuse. The hypotenuse is the side opposite the right angle (the little square) and it is the longest side. The side we want, d, is the one opposite the right angle, so it is the hypotenuse. We know the two shorter sides.Shorter sides: 15 km and 8 km.
- Write down Pythagoras’ theorem and put the numbers in. For a right-angled triangle, (shorter side)2 + (shorter side)2 = (hypotenuse)2.a2 + b2 = c2
152 + 82 = d2 - Square each short side. Squaring means multiplying the number by itself.152 = 15 × 15 = 225
82 = 8 × 8 = 64 - Add the two squares. This gives d2, the square of the hypotenuse.d2 = 225 + 64 = 289
- Take the square root. We want d, not d2. The square root undoes the squaring.d = √289 = 17 km
Check. The hypotenuse must be the longest side: 17 is bigger than 15 and 8. Also 152 + 82 = 289 = 172. ✓
Watch out. For the hypotenuse you add the squares. Do not stop at d2: the last step is the square root.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. To find the hypotenuse, square the two shorter sides, add them, then take the square root.
- Draw the right-angled triangle. The pole stands at 90° to the ground. The wire is the hypotenuse.Pole 7.5 m, distance on the ground 4 m, wire = x.
- Spot the hypotenuse. The hypotenuse is the side opposite the right angle (the little square) and it is the longest side. The side we want, x, is the one opposite the right angle, so it is the hypotenuse. We know the two shorter sides.Shorter sides: 4 m and 7.5 m.
- Write down Pythagoras’ theorem and put the numbers in. For a right-angled triangle, (shorter side)2 + (shorter side)2 = (hypotenuse)2.a2 + b2 = c2
42 + 7.52 = x2 - Square each short side. Squaring means multiplying the number by itself.42 = 4 × 4 = 16
7.52 = 7.5 × 7.5 = 56.25 - Add the two squares. This gives x2, the square of the hypotenuse.x2 = 16 + 56.25 = 72.25
- Take the square root. We want x, not x2. The square root undoes the squaring.x = √72.25 = 8.5 m
Check. The hypotenuse must be the longest side: 8.5 is bigger than 4 and 7.5. Also 42 + 7.52 = 72.25 = 8.52. ✓
Watch out. For the hypotenuse you add the squares. Do not stop at x2: the last step is the square root.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. To find a shorter side, square the hypotenuse and the known side, subtract, then take the square root.
- Draw the right-angled triangle. The ramp is the hypotenuse. The upright height and the flat ground make the two shorter sides.Hypotenuse 4.2 m. Height 0.35 m. The horizontal distance is d.
- Spot the hypotenuse. The hypotenuse is opposite the right angle and is the longest side. This time it is given, and we are looking for one of the two shorter sides.Hypotenuse = 4.2 m. The other known side = 0.35 m. The side we want is d.
- Write down Pythagoras’ theorem and put the numbers in. The hypotenuse is always the one on its own on the right-hand side: (shorter)2 + (shorter)2 = (hypotenuse)2.a2 + b2 = c2
d2 + 0.352 = 4.22 - Square the known numbers. Squaring means multiplying the number by itself.0.352 = 0.35 × 0.35 = 0.1225
4.22 = 4.2 × 4.2 = 17.64
d2 + 0.1225 = 17.64 - Subtract 0.1225 from both sides. We want d2 on its own. Taking 0.1225 away from both sides keeps the equation balanced.d2 = 17.64 − 0.1225 = 17.5175
- Take the square root. We want d, not d2.d = √17.5175 = 4.18539…
- Round to 2 d.p.. The question tells us how accurate to be.d = 4.19 m
Check. Put it back: 4.192 + 0.352 = 17.56 + 0.1225 = 17.68, which is close to 4.22 = 17.64. ✓ The answer is shorter than the hypotenuse 4.2.
Watch out. When you are finding a shorter side you subtract. A shorter side must always be smaller than the hypotenuse; if your answer is bigger, you added by mistake.
Key words. The hypotenuse is the longest side of a right-angled triangle; it is always opposite the right angle. Squaring a number means multiplying it by itself. The square root (√) undoes squaring.
Method. Let each side of the square be s. The diagonal is the hypotenuse of a right-angled triangle with both shorter sides equal to s.
- Write Pythagoras’ theorem for the triangle. The two shorter sides are both s and the hypotenuse is the diagonal, 10 cm.s2 + s2 = 102
- Collect the like terms. Two lots of s2 make 2s2.2s2 = 100
- Divide by 2. This leaves s2 on its own.s2 = 100 ÷ 2 = 50
- Square root and round. We want s, not s2.s = √50 = 7.07106…
s = 7.1 cm
Check. Put it back: 7.12 + 7.12 = 100.8, close to 102 = 100. ✓
Watch out. The two shorter sides are equal, so you can write one equation with only one letter.