GCSE Maths: Is it right-angled? Pythagorean triples and problems in context (Higher)
Part of the GCSE Maths pythagoras and trigonometry chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Is it right-angled? Pythagorean triples and problems in context
Pythagoras’ theorem works backwards too. If the squares of the two shorter sides add up to the square of the longest side, then the triangle must have a right angle. If they do not, it has no right angle. This gives us a test for a right angle using only lengths.
Whole numbers that fit the theorem are called a Pythagorean triple. The most famous is 3, 4, 5 (9 + 16 = 25). Others you should recognise are 5, 12, 13 and 8, 15, 17 and 7, 24, 25. If you multiply every number in a triple by the same amount you get another triple, so 6, 8, 10 and 30, 40, 50 also work. Builders use 3, 4, 5 to check that a corner is square: they measure 3 m along one wall and 4 m along the other and see if the gap between the ends is exactly 5 m.
If the sum of the squares of the two shorter sides is less than the square of the longest side, the biggest angle is more than 90° (obtuse). If it is more, the biggest angle is less than 90°. You do not need this to answer most questions, but it explains why the test says “not right-angled”.
Problems in context. Many real problems hide a right-angled triangle: a ladder against a wall, the diagonal of a rectangle, a ship that sails north then east, a wire holding up a pole. Follow these steps each time:
- Draw a sketch and mark the right angle.
- Mark the lengths you know, and call the one you want x.
- Decide whether x is the hypotenuse (the longest side, opposite the right angle) or a shorter side.
- Use Pythagoras: add the squares for the hypotenuse, subtract for a shorter side.
- Check that the answer is sensible, then write it with its unit.
Key ideas
- Test for a right angle: (short)2 + (short)2 = (longest)2.
- A Pythagorean triple is three whole numbers that fit: 3,4,5; 5,12,13; 8,15,17; 7,24,25.
- Multiplying a triple by the same number gives another triple.
- Always draw a sketch for a word problem and mark the right angle.
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. Square all three sides. If (short)2 + (short)2 equals (longest)2, the triangle has a right angle.
- Find the longest side. If the triangle is right-angled, the longest side must be the hypotenuse, so it goes on its own on one side of the equation.Sides in order: 7, 24, 25. Longest = 25 cm.
- Square the two shorter sides and add. This is the left-hand side of Pythagoras’ theorem.72 + 242 = 49 + 576 = 625
- Square the longest side. This is the right-hand side.252 = 625
- Compare the two answers. If they are equal, Pythagoras’ theorem works, so the triangle is right-angled. If not, it is not.625 = 625. They are equal, so the triangle is right-angled.
Check. Sides that fit Pythagoras’ theorem are called a Pythagorean triple. This triple cannot be made smaller, so it is a basic one. ✓
Watch out. Add the squares of the two shorter sides and compare with the square of the longest side. Do not add the longest side 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. Square all three sides. If (short)2 + (short)2 equals (longest)2, the triangle has a right angle.
- Find the longest side. If the triangle is right-angled, the longest side must be the hypotenuse, so it goes on its own on one side of the equation.Sides in order: 6, 7, 10. Longest = 10 cm.
- Square the two shorter sides and add. This is the left-hand side of Pythagoras’ theorem.62 + 72 = 36 + 49 = 85
- Square the longest side. This is the right-hand side.102 = 100
- Compare the two answers. If they are equal, Pythagoras’ theorem works, so the triangle is right-angled. If not, it is not.85 ≠ 100. They are not equal, so the triangle is not right-angled.
Check. If it were right-angled the longest side would be √(36 + 49) = 9.21954…, but it is 10. ✓ The longest side is longer than that, so the biggest angle is more than 90°.
Watch out. Add the squares of the two shorter sides and compare with the square of the longest side. Do not add the longest side 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 sketch. The wall is vertical and the ground is horizontal, so they make a right angle. The ladder is the hypotenuse because it is opposite that right angle.Ladder = 5.6 m (hypotenuse). Foot of ladder to the wall = 1.4 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 = 5.6 m. The other known side = 1.4 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 + 1.42 = 5.62 - Square the known numbers. Squaring means multiplying the number by itself.1.42 = 1.4 × 1.4 = 1.96
5.62 = 5.6 × 5.6 = 31.36
h2 + 1.96 = 31.36 - Subtract 1.96 from both sides. We want h2 on its own. Taking 1.96 away from both sides keeps the equation balanced.h2 = 31.36 − 1.96 = 29.4
- Take the square root. We want h, not h2.h = √29.4 = 5.42217…
- Round to 2 d.p.. The question tells us how accurate to be.h = 5.42 m
Check. Put it back: 5.422 + 1.42 = 29.38 + 1.96 = 31.34, which is close to 5.62 = 31.36. ✓ The answer is shorter than the hypotenuse 5.6.
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. A Pythagorean triple is a set of three whole numbers a, b, c with a2 + b2 = c2. To show that means to give every step of a proof.
Method. The longest side should be m2 + n2. Show that the two shorter sides squared and added equal its square, using algebra.
- Square the first short side. Use (p − q)2 = p2 − 2pq + q2 with p = m2 and q = n2.(m2 − n2)2 = m4 − 2m2n2 + n4
- Square the second short side. Square both the 2 and the mn.(2mn)2 = 4m2n2
- Add the two squares. Collect like terms: −2m2n2 + 4m2n2 = 2m2n2.m4 − 2m2n2 + n4 + 4m2n2 = m4 + 2m2n2 + n4
- Square the longest side. Use (p + q)2 = p2 + 2pq + q2.(m2 + n2)2 = m4 + 2m2n2 + n4
- Compare. The two results are the same, so Pythagoras’ theorem holds and the triangle has a right angle.(m2 − n2)2 + (2mn)2 = (m2 + n2)2
- Make a triple with m = 3 and n = 2. Substitute into the three expressions.32 − 22 = 9 − 4 = 5
2 × 3 × 2 = 12
32 + 22 = 9 + 4 = 13
The triple is 5, 12, 13.
Check. 52 + 122 = 25 + 144 = 169 = 132. ✓
Watch out. Expand brackets carefully: (m2 − n2)2 is not m4 − n4.
Watch out
When testing for a right angle, put the longest side on its own. A common mistake is to add the square of the longest side to another one. Also, “close enough” is not good enough: 6, 7, 10 has 36 + 49 = 85, which is not 100, so there is no right angle.Exercise 20C Right-angled triangles in problems
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. Square all three sides. If (short)2 + (short)2 equals (longest)2, the triangle has a right angle.
- Find the longest side. If the triangle is right-angled, the longest side must be the hypotenuse, so it goes on its own on one side of the equation.Sides in order: 3, 4, 5. Longest = 5 cm.
- Square the two shorter sides and add. This is the left-hand side of Pythagoras’ theorem.32 + 42 = 9 + 16 = 25
- Square the longest side. This is the right-hand side.52 = 25
- Compare the two answers. If they are equal, Pythagoras’ theorem works, so the triangle is right-angled. If not, it is not.25 = 25. They are equal, so the triangle is right-angled.
Check. Sides that fit Pythagoras’ theorem are called a Pythagorean triple. This triple cannot be made smaller, so it is a basic one. ✓
Watch out. Add the squares of the two shorter sides and compare with the square of the longest side. Do not add the longest side 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. Square all three sides. If (short)2 + (short)2 equals (longest)2, the triangle has a right angle.
- Find the longest side. If the triangle is right-angled, the longest side must be the hypotenuse, so it goes on its own on one side of the equation.Sides in order: 5, 12, 13. Longest = 13 cm.
- Square the two shorter sides and add. This is the left-hand side of Pythagoras’ theorem.52 + 122 = 25 + 144 = 169
- Square the longest side. This is the right-hand side.132 = 169
- Compare the two answers. If they are equal, Pythagoras’ theorem works, so the triangle is right-angled. If not, it is not.169 = 169. They are equal, so the triangle is right-angled.
Check. Sides that fit Pythagoras’ theorem are called a Pythagorean triple. This triple cannot be made smaller, so it is a basic one. ✓
Watch out. Add the squares of the two shorter sides and compare with the square of the longest side. Do not add the longest side 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. Square all three sides. If (short)2 + (short)2 equals (longest)2, the triangle has a right angle.
- Find the longest side. If the triangle is right-angled, the longest side must be the hypotenuse, so it goes on its own on one side of the equation.Sides in order: 6, 8, 11. Longest = 11 cm.
- Square the two shorter sides and add. This is the left-hand side of Pythagoras’ theorem.62 + 82 = 36 + 64 = 100
- Square the longest side. This is the right-hand side.112 = 121
- Compare the two answers. If they are equal, Pythagoras’ theorem works, so the triangle is right-angled. If not, it is not.100 ≠ 121. They are not equal, so the triangle is not right-angled.
Check. If it were right-angled the longest side would be √(36 + 64) = 10, but it is 11. ✓ The longest side is longer than that, so the biggest angle is more than 90°.
Watch out. Add the squares of the two shorter sides and compare with the square of the longest side. Do not add the longest side 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. Square all three sides. If (short)2 + (short)2 equals (longest)2, the triangle has a right angle.
- Find the longest side. If the triangle is right-angled, the longest side must be the hypotenuse, so it goes on its own on one side of the equation.Sides in order: 9, 12, 15. Longest = 15 cm.
- Square the two shorter sides and add. This is the left-hand side of Pythagoras’ theorem.92 + 122 = 81 + 144 = 225
- Square the longest side. This is the right-hand side.152 = 225
- Compare the two answers. If they are equal, Pythagoras’ theorem works, so the triangle is right-angled. If not, it is not.225 = 225. They are equal, so the triangle is right-angled.
Check. Sides that fit Pythagoras’ theorem are called a Pythagorean triple. Here 3, 4, 5 is a smaller triple and every side has been multiplied by 3. ✓
Watch out. Add the squares of the two shorter sides and compare with the square of the longest side. Do not add the longest side 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. Square all three sides. If (short)2 + (short)2 equals (longest)2, the triangle has a right angle.
- Find the longest side. If the triangle is right-angled, the longest side must be the hypotenuse, so it goes on its own on one side of the equation.Sides in order: 7, 8, 12. Longest = 12 cm.
- Square the two shorter sides and add. This is the left-hand side of Pythagoras’ theorem.72 + 82 = 49 + 64 = 113
- Square the longest side. This is the right-hand side.122 = 144
- Compare the two answers. If they are equal, Pythagoras’ theorem works, so the triangle is right-angled. If not, it is not.113 ≠ 144. They are not equal, so the triangle is not right-angled.
Check. If it were right-angled the longest side would be √(49 + 64) = 10.63014…, but it is 12. ✓ The longest side is longer than that, so the biggest angle is more than 90°.
Watch out. Add the squares of the two shorter sides and compare with the square of the longest side. Do not add the longest side 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. Square all three sides. If (short)2 + (short)2 equals (longest)2, the triangle has a right angle.
- Find the longest side. If the triangle is right-angled, the longest side must be the hypotenuse, so it goes on its own on one side of the equation.Sides in order: 20, 21, 29. Longest = 29 cm.
- Square the two shorter sides and add. This is the left-hand side of Pythagoras’ theorem.202 + 212 = 400 + 441 = 841
- Square the longest side. This is the right-hand side.292 = 841
- Compare the two answers. If they are equal, Pythagoras’ theorem works, so the triangle is right-angled. If not, it is not.841 = 841. They are equal, so the triangle is right-angled.
Check. Sides that fit Pythagoras’ theorem are called a Pythagorean triple. This triple cannot be made smaller, so it is a basic one. ✓
Watch out. Add the squares of the two shorter sides and compare with the square of the longest side. Do not add the longest side 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: 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 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 = 9 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 + 92 = 412 - Square the known numbers. Squaring means multiplying the number by itself.92 = 9 × 9 = 81
412 = 41 × 41 = 1681
x2 + 81 = 1681 - Subtract 81 from both sides. We want x2 on its own. Taking 81 away from both sides keeps the equation balanced.x2 = 1681 − 81 = 1600
- Take the square root. We want x, not x2.x = √1600 = 40 cm
Check. Put it back: 402 + 92 = 1600 + 81 = 1681 = 412. ✓ And 40 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.
- 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 = 29 cm. The other known side = 21 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 + 212 = 292 - Square the known numbers. Squaring means multiplying the number by itself.212 = 21 × 21 = 441
292 = 29 × 29 = 841
x2 + 441 = 841 - Subtract 441 from both sides. We want x2 on its own. Taking 441 away from both sides keeps the equation balanced.x2 = 841 − 441 = 400
- Take the square root. We want x, not x2.x = √400 = 20 cm
Check. Put it back: 202 + 212 = 400 + 441 = 841 = 292. ✓ And 20 is shorter than the hypotenuse 29.
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: 12 cm and 16 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 + 162 = x2 - Square each short side. Squaring means multiplying the number by itself.122 = 12 × 12 = 144
162 = 16 × 16 = 256 - Add the two squares. This gives x2, the square of the hypotenuse.x2 = 144 + 256 = 400
- Take the square root. We want x, not x2. The square root undoes the squaring.x = √400 = 20 cm
Check. The hypotenuse must be the longest side: 20 is bigger than 12 and 16. Also 122 + 162 = 400 = 202. ✓
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. Square all three sides. If (short)2 + (short)2 equals (longest)2, the triangle has a right angle.
- Find the longest side. If the triangle is right-angled, the longest side must be the hypotenuse, so it goes on its own on one side of the equation.Sides in order: 0.9, 1.2, 1.5. Longest = 1.5 m.
- Square the two shorter sides and add. This is the left-hand side of Pythagoras’ theorem.0.92 + 1.22 = 0.81 + 1.44 = 2.25
- Square the longest side. This is the right-hand side.1.52 = 2.25
- Compare the two answers. If they are equal, Pythagoras’ theorem works, so the triangle is right-angled. If not, it is not.2.25 = 2.25. They are equal, so the triangle is right-angled.
Check. Sides that fit Pythagoras’ theorem are called a Pythagorean triple. This triple cannot be made smaller, so it is a basic one. ✓
Watch out. Add the squares of the two shorter sides and compare with the square of the longest side. Do not add the longest side 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.
- Draw a sketch. The wall and ground make a right angle, so the ladder is the hypotenuse.The ladder reaches 5.2 m up the wall (a shorter side) and its foot is 1.5 m from the wall (a shorter side). The ladder length is L.
- 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, L, is the one opposite the right angle, so it is the hypotenuse. We know the two shorter sides.Shorter sides: 5.2 m and 1.5 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
5.22 + 1.52 = L2 - Square each short side. Squaring means multiplying the number by itself.5.22 = 5.2 × 5.2 = 27.04
1.52 = 1.5 × 1.5 = 2.25 - Add the two squares. This gives L2, the square of the hypotenuse.L2 = 27.04 + 2.25 = 29.29
- Take the square root. We want L, not L2. The square root undoes the squaring. Use the √ button on the calculator.L = √29.29 = 5.41202……
- Round to 2 d.p.. The question tells us how accurate to be.L = 5.41 m
Check. Square the answer: 5.412 = 29.27, which is close to 29.29. The hypotenuse 5.41 is longer than both 5.2 and 1.5. ✓
Watch out. For the hypotenuse you add the squares. Do not stop at L2: 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 journey. West and south are at right angles, so the journey and the straight line back to the start make a right-angled triangle.Shorter sides 24 km and 10 km. The distance d from the start 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: 24 km and 10 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
242 + 102 = d2 - 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 d2, the square of the hypotenuse.d2 = 576 + 100 = 676
- Take the square root. We want d, not d2. The square root undoes the squaring.d = √676 = 26 km
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 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.
- Use the properties of a rhombus. The diagonals of a rhombus cross at right angles and cut each other in half. Each side of the rhombus is the hypotenuse of a right-angled triangle whose shorter sides are half of each diagonal.Half of 16 cm = 8 cm and half of 12 cm = 6 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: 8 cm and 6 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 + 62 = x2 - Square each short side. Squaring means multiplying the number by itself.82 = 8 × 8 = 64
62 = 6 × 6 = 36 - Add the two squares. This gives x2, the square of the hypotenuse.x2 = 64 + 36 = 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 8 and 6. Also 82 + 62 = 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.
- Work out how far each cyclist rides. Distance = speed × time.Cyclist 1: 12 × 2 = 24 km north
Cyclist 2: 5 × 2 = 10 km east - Draw the right-angled triangle. North and east are at 90° to each other, so the distance between the cyclists is the hypotenuse.Shorter sides 24 km and 10 km.
- 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: 24 km and 10 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
242 + 102 = d2 - 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 d2, the square of the hypotenuse.d2 = 576 + 100 = 676
- Take the square root. We want d, not d2. The square root undoes the squaring.d = √676 = 26 km
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 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. Square all three sides. If (short)2 + (short)2 equals (longest)2, the triangle has a right angle.
- Find the longest side. If the triangle is right-angled, the longest side must be the hypotenuse, so it goes on its own on one side of the equation.Sides in order: 0.9, 2.1, 2.3. Longest = 2.3 m.
- Square the two shorter sides and add. This is the left-hand side of Pythagoras’ theorem.0.92 + 2.12 = 0.81 + 4.41 = 5.22
- Square the longest side. This is the right-hand side.2.32 = 5.29
- Compare the two answers. If they are equal, Pythagoras’ theorem works, so the triangle is right-angled. If not, it is not.5.22 ≠ 5.29. They are not equal, so the triangle is not right-angled.
Check. If it were right-angled the longest side would be √(0.81 + 4.41) = 2.28473…, but it is 2.3. ✓ The longest side is longer than that, so the biggest angle is more than 90°.
Watch out. Add the squares of the two shorter sides and compare with the square of the longest side. Do not add the longest side 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 sketch. The pole is upright and the ground is flat, so they make a right angle. The wire is the hypotenuse.Wire = 10 m (hypotenuse). Pole = 6 m. The distance on the ground 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 = 10 m. The other known side = 6 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 + 62 = 102 - Square the known numbers. Squaring means multiplying the number by itself.62 = 6 × 6 = 36
102 = 10 × 10 = 100
d2 + 36 = 100 - Subtract 36 from both sides. We want d2 on its own. Taking 36 away from both sides keeps the equation balanced.d2 = 100 − 36 = 64
- Take the square root. We want d, not d2.d = √64 = 8 m
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. The test needs the squares of the sides, not the sides themselves.
- Say what is wrong. Pythagoras’ theorem is about a2 + b2 = c2. Adding the plain lengths does not test for a right angle (the two shorter sides added are always longer than the third side, even in a right-angled triangle).Raj did 8 + 15, which is not the test.
- Use the correct test. Square the two shorter sides and add. Then square the longest side.82 + 152 = 64 + 225 = 289
162 = 256 - Compare. The numbers must be equal for a right angle.289 ≠ 256, so the triangle is not right-angled.
Check. 8, 15, 17 is the right-angled triangle, because 64 + 225 = 289 = 172. Changing 17 to 16 spoils it. ✓
Watch out. Raj reaches the right conclusion with the wrong method. In an exam you would get no marks for the method.
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. Check the doubled triangle with the test for a right angle, then use algebra to see why it always works.
- Check the doubled triangle. Double 5, 12 and 13 to get 10, 24 and 26. Test them.102 + 242 = 100 + 576 = 676
262 = 676
They are equal, so the doubled triangle is right-angled. - Explain for any triangle. Suppose a2 + b2 = c2. After doubling, the sides are 2a, 2b and 2c. Squaring doubles into 4 times the square.(2a)2 + (2b)2 = 4a2 + 4b2 = 4(a2 + b2) = 4c2 = (2c)2
- Write the conclusion. Both sides of the equation are multiplied by 4, so it is still true.The doubled triangle is always right-angled. The same works for any scale factor.
Check. The new triangle is an enlargement, so it has the same angles. The right angle stays a right angle. ✓
Watch out. In a “show that” question write a sentence at the end that says what you have shown.
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 longest side (x + 8) is the hypotenuse. Put the three sides into Pythagoras’ theorem, then solve the quadratic equation.
- Write Pythagoras’ theorem. The hypotenuse is x + 8.x2 + (x + 7)2 = (x + 8)2
- Expand the brackets. (x + 7)2 = x2 + 14x + 49 and (x + 8)2 = x2 + 16x + 64.x2 + x2 + 14x + 49 = x2 + 16x + 64
- Collect everything on the left. Subtract x2, 16x and 64 from both sides.x2 − 2x − 15 = 0
- Factorise. We need two numbers that multiply to −15 and add to −2: −5 and +3.(x − 5)(x + 3) = 0
- Solve and reject. A length cannot be negative.x = 5 or x = −3. So x = 5.
Check. Sides 5, 12, 13: 25 + 144 = 169. ✓
Watch out. Always reject any solution that gives a negative length.