GCSE Maths: Pythagoras with coordinates, isosceles and equilateral triangles (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)
Pythagoras with coordinates, isosceles and equilateral triangles
Distance between two points. To find the distance between two points on a grid, imagine a right-angled triangle between them. The straight line joining the points is the hypotenuse. The horizontal step is the difference of the x-coordinates and the vertical step is the difference of the y-coordinates. Then use Pythagoras. A minus sign in a step does not matter, because squaring turns it into a positive number.
Isosceles triangles. An isosceles triangle has two equal sides, and the angles at the base are equal. If you draw the line of symmetry from the top down to the base, it hits the base at 90° and cuts it exactly in half. That makes two right-angled triangles that are mirror images of each other. Their hypotenuse is one of the equal sides, one shorter side is half the base and the other shorter side is the height of the triangle. Once you know the height you can find the area: ½ × base × height.
Equilateral triangles. An equilateral triangle has three equal sides and three 60° angles. It is a special isosceles triangle, so the same trick works. For a side of length s the half-base is s ÷ 2, so the height h satisfies h2 + (s ÷ 2)2 = s2.
Higher tier: you may be asked for an exact answer. Do not round. Leave square roots in the answer and simplify them, for example √52 = √4 × √13 = 2√13.
Key ideas
- Distance between points: (x step)2 + (y step)2 = distance2.
- An isosceles triangle splits into two right-angled triangles: use half the base.
- Area of the triangle = ½ × whole base × height.
- Equilateral triangle: height2 = side2 − (½ side)2.
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 distance between two points is the hypotenuse of a right-angled triangle whose shorter sides are the horizontal and vertical steps between them.
- Draw a right-angled triangle between the two points. The line from one point to the other is the hypotenuse. The two shorter sides are the horizontal step and the vertical step.Horizontal step across, vertical step up or down.
- Find the horizontal and vertical steps. Subtract the x-coordinates, then subtract the y-coordinates. Only the size of each step matters, so ignore any minus sign.Horizontal: 7 − 1 = 6, so the step is 6
Vertical: 10 − 2 = 8, so the step is 8 - Use Pythagoras’ theorem. The distance d is the hypotenuse.d2 = 62 + 82 = 36 + 64 = 100
- Take the square root. We want d, not d2.d = √100 = 10
Check. 6, 8, 10 fits a Pythagorean triple pattern: 36 + 64 = 100. ✓
Watch out. Subtract the coordinates in the same order for x and y. A negative step is fine: squaring removes the minus sign.
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 distance between two points is the hypotenuse of a right-angled triangle whose shorter sides are the horizontal and vertical steps between them.
- Draw a right-angled triangle between the two points. The line from one point to the other is the hypotenuse. The two shorter sides are the horizontal step and the vertical step.Horizontal step across, vertical step up or down.
- Find the horizontal and vertical steps. Subtract the x-coordinates, then subtract the y-coordinates. Only the size of each step matters, so ignore any minus sign.Horizontal: 5 − (−3) = 8, so the step is 8
Vertical: −2 − 4 = −6, so the step is 6 - Use Pythagoras’ theorem. The distance d is the hypotenuse.d2 = 82 + 62 = 64 + 36 = 100
- Take the square root. We want d, not d2.d = √100 = 10
Check. 8, 6, 10 fits a Pythagorean triple pattern: 64 + 36 = 100. ✓
Watch out. Subtract the coordinates in the same order for x and y. A negative step is fine: squaring removes the minus sign.
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. An isosceles triangle has two equal sides.
Method. Split the isosceles triangle down the middle into two right-angled triangles, then use Pythagoras’ theorem.
- Draw the line of symmetry from the top to the middle of the base. In an isosceles triangle this line is the perpendicular height. It cuts the base in half and splits the triangle into two identical right-angled triangles.Half the base = 10 ÷ 2 = 5 cm
- Pick one right-angled triangle. Its hypotenuse is the slanting side, 13 cm. Its shorter sides are the half-base and the height h.h2 + 52 = 132
- Square and subtract. We are finding a shorter side, so we subtract.h2 = 169 − 25 = 144
- Take the square root. The height is h, not h2.h = √144 = 12 cm
- Use area of a triangle = ½ × base × height. The base is the whole base, not the half. Use the exact height from the calculator to avoid rounding errors.Area = ½ × 10 × 12 = 60
- Write the area with its units. Area is measured in square units. The answer is exact, so no rounding is needed.Area = 60 cm2
Check. Another way: two small triangles each have area ½ × 5 × 12 = 30.00; doubling gives 60.00, matching 60. ✓
Watch out. Use half the base in Pythagoras’ theorem, but the whole base in the area formula.
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. An isosceles triangle has two equal sides.
Method. Split the isosceles triangle down the middle into two right-angled triangles, then use Pythagoras’ theorem.
- Draw the line of symmetry from the top to the middle of the base. In an isosceles triangle this line is the perpendicular height. It cuts the base in half and splits the triangle into two identical right-angled triangles.Half the base = 8 ÷ 2 = 4 cm
- Pick one right-angled triangle. Its hypotenuse is the slanting side, 8 cm. Its shorter sides are the half-base and the height h.h2 + 42 = 82
- Square and subtract. We are finding a shorter side, so we subtract.h2 = 64 − 16 = 48
- Take the square root. The height is h, not h2.h = √48 = 6.92820…
h = 6.93 cm
Check. The height (6.93) must be shorter than the slanting side (8). Also 6.932 + 42 = 64.02, close to 64. ✓
Watch out. Use half the base in Pythagoras’ theorem, but the whole base in the area formula.
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 distance between two points is the hypotenuse of a right-angled triangle whose shorter sides are the horizontal and vertical steps between them.
- Draw a right-angled triangle between the two points. The line from one point to the other is the hypotenuse. The two shorter sides are the horizontal step and the vertical step.Horizontal step across, vertical step up or down.
- Find the horizontal and vertical steps. Subtract the x-coordinates, then subtract the y-coordinates. Only the size of each step matters, so ignore any minus sign.Horizontal: 4 − (−2) = 6, so the step is 6
Vertical: 5 − 1 = 4, so the step is 4 - Use Pythagoras’ theorem. The distance d is the hypotenuse.d2 = 62 + 42 = 36 + 16 = 52
- Take the square root and leave it exact. An exact answer is wanted, so we simplify the surd.d = √52
Find the biggest square number that divides 52: 52 = 4 × 13
√52 = √4 × √13 = 2√13
Check. 2√13 = 7.21110… on a calculator, and the line looks about that long on the grid. ✓
Watch out. Subtract the coordinates in the same order for x and y. A negative step is fine: squaring removes the minus sign.
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. Split the triangle in half down its line of symmetry. We get a right-angled triangle with hypotenuse 2a, shorter sides a and h.
- Write Pythagoras for the half-triangle. The hypotenuse is the full side 2a. The base of the half-triangle is half of 2a, which is a.h2 + a2 = (2a)2
- Expand and rearrange. (2a)2 = 4a2. Subtract a2 from both sides.h2 = 4a2 − a2 = 3a2
- Take the square root. √(3a2) = √3 × √(a2) = √3 × a, because a is a positive length.h = a√3
- Find the area. Area = ½ × whole base × height.Area = ½ × 2a × a√3 = a2√3
Check. Test with a = 4: side 8, height = 4√3 = 6.93 (matches the earlier example). ✓
Watch out. The base of the half-triangle is a (half of 2a), not 2a.
Watch out
For an isosceles triangle, use half the base in Pythagoras’ theorem, but the whole base in the area formula ½ × base × height. Mixing these up is the most common mistake.Exercise 20D Coordinates, isosceles and equilateral triangles
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. The distance between two points is the hypotenuse of a right-angled triangle whose shorter sides are the horizontal and vertical steps between them.
- Draw a right-angled triangle between the two points. The line from one point to the other is the hypotenuse. The two shorter sides are the horizontal step and the vertical step.Horizontal step across, vertical step up or down.
- Find the horizontal and vertical steps. Subtract the x-coordinates, then subtract the y-coordinates. Only the size of each step matters, so ignore any minus sign.Horizontal: 4 − 1 = 3, so the step is 3
Vertical: 6 − 2 = 4, so the step is 4 - Use Pythagoras’ theorem. The distance d is the hypotenuse.d2 = 32 + 42 = 9 + 16 = 25
- Take the square root. We want d, not d2.d = √25 = 5
Check. 3, 4, 5 fits a Pythagorean triple pattern: 9 + 16 = 25. ✓
Watch out. Subtract the coordinates in the same order for x and y. A negative step is fine: squaring removes the minus sign.
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 distance between two points is the hypotenuse of a right-angled triangle whose shorter sides are the horizontal and vertical steps between them.
- Draw a right-angled triangle between the two points. The line from one point to the other is the hypotenuse. The two shorter sides are the horizontal step and the vertical step.Horizontal step across, vertical step up or down.
- Find the horizontal and vertical steps. Subtract the x-coordinates, then subtract the y-coordinates. Only the size of each step matters, so ignore any minus sign.Horizontal: 5 − 0 = 5, so the step is 5
Vertical: −9 − 3 = −12, so the step is 12 - Use Pythagoras’ theorem. The distance d is the hypotenuse.d2 = 52 + 122 = 25 + 144 = 169
- Take the square root. We want d, not d2.d = √169 = 13
Check. 5, 12, 13 fits a Pythagorean triple pattern: 25 + 144 = 169. ✓
Watch out. Subtract the coordinates in the same order for x and y. A negative step is fine: squaring removes the minus sign.
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 distance between two points is the hypotenuse of a right-angled triangle whose shorter sides are the horizontal and vertical steps between them.
- Draw a right-angled triangle between the two points. The line from one point to the other is the hypotenuse. The two shorter sides are the horizontal step and the vertical step.Horizontal step across, vertical step up or down.
- Find the horizontal and vertical steps. Subtract the x-coordinates, then subtract the y-coordinates. Only the size of each step matters, so ignore any minus sign.Horizontal: 3 − (−4) = 7, so the step is 7
Vertical: 5 − 2 = 3, so the step is 3 - Use Pythagoras’ theorem. The distance d is the hypotenuse.d2 = 72 + 32 = 49 + 9 = 58
- Take the square root and round. We want d, not d2.d = √58 = 7.61577…
d = 7.6 (to 1 d.p.)
Check. Square the answer: 7.62 = 57.76, close to 58. ✓
Watch out. Subtract the coordinates in the same order for x and y. A negative step is fine: squaring removes the minus sign.
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 distance between two points is the hypotenuse of a right-angled triangle whose shorter sides are the horizontal and vertical steps between them.
- Draw a right-angled triangle between the two points. The line from one point to the other is the hypotenuse. The two shorter sides are the horizontal step and the vertical step.Horizontal step across, vertical step up or down.
- Find the horizontal and vertical steps. Subtract the x-coordinates, then subtract the y-coordinates. Only the size of each step matters, so ignore any minus sign.Horizontal: 2 − (−6) = 8, so the step is 8
Vertical: 4 − (−1) = 5, so the step is 5 - Use Pythagoras’ theorem. The distance d is the hypotenuse.d2 = 82 + 52 = 64 + 25 = 89
- Take the square root and round. We want d, not d2.d = √89 = 9.43398…
d = 9.4 (to 1 d.p.)
Check. Square the answer: 9.42 = 88.36, close to 89. ✓
Watch out. Subtract the coordinates in the same order for x and y. A negative step is fine: squaring removes the minus sign.
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 distance between two points is the hypotenuse of a right-angled triangle whose shorter sides are the horizontal and vertical steps between them.
- Draw a right-angled triangle between the two points. The line from one point to the other is the hypotenuse. The two shorter sides are the horizontal step and the vertical step.Horizontal step across, vertical step up or down.
- Find the horizontal and vertical steps. Subtract the x-coordinates, then subtract the y-coordinates. Only the size of each step matters, so ignore any minus sign.Horizontal: 14 − 2 = 12, so the step is 12
Vertical: 8 − 3 = 5, so the step is 5 - Use Pythagoras’ theorem. The distance d is the hypotenuse.d2 = 122 + 52 = 144 + 25 = 169
- Take the square root. We want d, not d2.d = √169 = 13
Check. 12, 5, 13 fits a Pythagorean triple pattern: 144 + 25 = 169. ✓
Watch out. Subtract the coordinates in the same order for x and y. A negative step is fine: squaring removes the minus sign.
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. A triangle is isosceles if two of its sides are equal. Work out the three side lengths using Pythagoras’ theorem.
- Find AB. The steps from A(2, 1) to B(6, 4) are 4 across and 3 up.AB2 = 42 + 32 = 16 + 9 = 25, so AB = 5
- Find BC. The steps from B(6, 4) to C(2, 7) are 4 across (left) and 3 up.BC2 = 42 + 32 = 25, so BC = 5
- Find AC. A(2, 1) and C(2, 7) are straight above each other, so the distance is just the difference of the y-values.AC = 7 − 1 = 6
- Compare. Two sides are the same length.AB = BC = 5 and AC = 6, so the triangle is isosceles.
Check. The midpoint of AC is (2, 4). It has the same y-coordinate as B(6, 4), so B is level with the middle of AC, and the triangle is symmetrical. ✓
Watch out. Check all three sides. Two equal sides is enough; the third can be different.
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. An isosceles triangle has two equal sides.
Method. Split the isosceles triangle down the middle into two right-angled triangles, then use Pythagoras’ theorem.
- Draw the line of symmetry from the top to the middle of the base. In an isosceles triangle this line is the perpendicular height. It cuts the base in half and splits the triangle into two identical right-angled triangles.Half the base = 12 ÷ 2 = 6 cm
- Pick one right-angled triangle. Its hypotenuse is the slanting side, 10 cm. Its shorter sides are the half-base and the height h.h2 + 62 = 102
- Square and subtract. We are finding a shorter side, so we subtract.h2 = 100 − 36 = 64
- Take the square root. The height is h, not h2.h = √64 = 8 cm
Check. The height (8) must be shorter than the slanting side (10). Also 82 + 62 = 100.00, close to 100. ✓
Watch out. Use half the base in Pythagoras’ theorem, but the whole base in the area formula.
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. An isosceles triangle has two equal sides.
Method. Split the isosceles triangle down the middle into two right-angled triangles, then use Pythagoras’ theorem.
- Draw the line of symmetry from the top to the middle of the base. In an isosceles triangle this line is the perpendicular height. It cuts the base in half and splits the triangle into two identical right-angled triangles.Half the base = 10 ÷ 2 = 5 cm
- Pick one right-angled triangle. Its hypotenuse is the slanting side, 13 cm. Its shorter sides are the half-base and the height h.h2 + 52 = 132
- Square and subtract. We are finding a shorter side, so we subtract.h2 = 169 − 25 = 144
- Take the square root. The height is h, not h2.h = √144 = 12 cm
Check. The height (12) must be shorter than the slanting side (13). Also 122 + 52 = 169.00, close to 169. ✓
Watch out. Use half the base in Pythagoras’ theorem, but the whole base in the area formula.
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. An isosceles triangle has two equal sides.
Method. Split the isosceles triangle down the middle into two right-angled triangles, then use Pythagoras’ theorem.
- Draw the line of symmetry from the top to the middle of the base. In an isosceles triangle this line is the perpendicular height. It cuts the base in half and splits the triangle into two identical right-angled triangles.Half the base = 16 ÷ 2 = 8 cm
- Pick one right-angled triangle. Its hypotenuse is the slanting side, 17 cm. Its shorter sides are the half-base and the height h.h2 + 82 = 172
- Square and subtract. We are finding a shorter side, so we subtract.h2 = 289 − 64 = 225
- Take the square root. The height is h, not h2.h = √225 = 15 cm
Check. The height (15) must be shorter than the slanting side (17). Also 152 + 82 = 289.00, close to 289. ✓
Watch out. Use half the base in Pythagoras’ theorem, but the whole base in the area formula.
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. An isosceles triangle has two equal sides.
Method. Split the isosceles triangle down the middle into two right-angled triangles, then use Pythagoras’ theorem.
- Draw the line of symmetry from the top to the middle of the base. In an isosceles triangle this line is the perpendicular height. It cuts the base in half and splits the triangle into two identical right-angled triangles.Half the base = 5 ÷ 2 = 2.5 cm
- Pick one right-angled triangle. Its hypotenuse is the slanting side, 6.5 cm. Its shorter sides are the half-base and the height h.h2 + 2.52 = 6.52
- Square and subtract. We are finding a shorter side, so we subtract.h2 = 42.25 − 6.25 = 36
- Take the square root. The height is h, not h2.h = √36 = 6 cm
Check. The height (6) must be shorter than the slanting side (6.5). Also 62 + 2.52 = 42.25, close to 42.25. ✓
Watch out. Use half the base in Pythagoras’ theorem, but the whole base in the area formula.
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. An isosceles triangle has two equal sides.
Method. Split the isosceles triangle down the middle into two right-angled triangles, then use Pythagoras’ theorem.
- Draw the line of symmetry from the top to the middle of the base. In an isosceles triangle this line is the perpendicular height. It cuts the base in half and splits the triangle into two identical right-angled triangles.Half the base = 12 ÷ 2 = 6 cm
- Pick one right-angled triangle. Its hypotenuse is the slanting side, 10 cm. Its shorter sides are the half-base and the height h.h2 + 62 = 102
- Square and subtract. We are finding a shorter side, so we subtract.h2 = 100 − 36 = 64
- Take the square root. The height is h, not h2.h = √64 = 8 cm
- Use area of a triangle = ½ × base × height. The base is the whole base, not the half. Use the exact height from the calculator to avoid rounding errors.Area = ½ × 12 × 8 = 48
- Write the area with its units. Area is measured in square units. The answer is exact, so no rounding is needed.Area = 48 cm2
Check. Another way: two small triangles each have area ½ × 6 × 8 = 24.00; doubling gives 48.00, matching 48. ✓
Watch out. Use half the base in Pythagoras’ theorem, but the whole base in the area formula.
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. An isosceles triangle has two equal sides.
Method. Split the isosceles triangle down the middle into two right-angled triangles, then use Pythagoras’ theorem.
- Draw the line of symmetry from the top to the middle of the base. In an isosceles triangle this line is the perpendicular height. It cuts the base in half and splits the triangle into two identical right-angled triangles.Half the base = 10 ÷ 2 = 5 cm
- Pick one right-angled triangle. Its hypotenuse is the slanting side, 10 cm. Its shorter sides are the half-base and the height h.h2 + 52 = 102
- Square and subtract. We are finding a shorter side, so we subtract.h2 = 100 − 25 = 75
- Take the square root. The height is h, not h2.h = √75 = 8.66025…
h = 8.66 cm
Check. The height (8.66) must be shorter than the slanting side (10). Also 8.662 + 52 = 100.00, close to 100. ✓
Watch out. Use half the base in Pythagoras’ theorem, but the whole base in the area formula.
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. An isosceles triangle has two equal sides.
Method. Split the isosceles triangle down the middle into two right-angled triangles, then use Pythagoras’ theorem.
- Draw the line of symmetry from the top to the middle of the base. In an isosceles triangle this line is the perpendicular height. It cuts the base in half and splits the triangle into two identical right-angled triangles.Half the base = 9.6 ÷ 2 = 4.8 m
- Pick one right-angled triangle. Its hypotenuse is the slanting side, 5.2 m. Its shorter sides are the half-base and the height h.h2 + 4.82 = 5.22
- Square and subtract. We are finding a shorter side, so we subtract.h2 = 27.04 − 23.04 = 4
- Take the square root. The height is h, not h2.h = √4 = 2 m
Check. The height (2) must be shorter than the slanting side (5.2). Also 22 + 4.82 = 27.04, close to 27.04. ✓
Watch out. Use half the base in Pythagoras’ theorem, but the whole base in the area formula.
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. An isosceles triangle has two equal sides.
Method. Split the isosceles triangle down the middle into two right-angled triangles, then use Pythagoras’ theorem.
- Draw the line of symmetry from the top to the middle of the base. In an isosceles triangle this line is the perpendicular height. It cuts the base in half and splits the triangle into two identical right-angled triangles.Half the base = 2.4 ÷ 2 = 1.2 m
- Pick one right-angled triangle. Its hypotenuse is the slanting side, 1.9 m. Its shorter sides are the half-base and the height h.h2 + 1.22 = 1.92
- Square and subtract. We are finding a shorter side, so we subtract.h2 = 3.61 − 1.44 = 2.17
- Take the square root. The height is h, not h2.h = √2.17 = 1.47309…
h = 1.47 m
Check. The height (1.47) must be shorter than the slanting side (1.9). Also 1.472 + 1.22 = 3.60, close to 3.61. ✓
Watch out. Use half the base in Pythagoras’ theorem, but the whole base in the area formula.
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. An isosceles triangle has two equal sides.
Method. Split the isosceles triangle down the middle into two right-angled triangles, then use Pythagoras’ theorem.
- Draw the line of symmetry from the top to the middle of the base. In an isosceles triangle this line is the perpendicular height. It cuts the base in half and splits the triangle into two identical right-angled triangles.Half the base = 50 ÷ 2 = 25 cm
- Pick one right-angled triangle. Its hypotenuse is the slanting side, 50 cm. Its shorter sides are the half-base and the height h.h2 + 252 = 502
- Square and subtract. We are finding a shorter side, so we subtract.h2 = 2500 − 625 = 1875
- Take the square root. The height is h, not h2.h = √1875 = 43.30127…
h = 43.30 cm (keep extra digits for now) - Use area of a triangle = ½ × base × height. The base is the whole base, not the half. Use the exact height from the calculator to avoid rounding errors.Area = ½ × 50 × 43.30127… = 1082.53175…
- Round to 1 d.p. Give the units for area: square units.Area = 1082.5 cm2
Check. Another way: two small triangles each have area ½ × 25 × 43.30 = 541.25; doubling gives 1082.50, matching 1082.5. ✓
Watch out. Use half the base in Pythagoras’ theorem, but the whole base in the area formula.
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 distance is the hypotenuse of a right-angled triangle, so we square the steps, add and take the root. We do not just add the steps.
- Find the steps. Subtract the coordinates.Across: 4 − 1 = 3. Up: 6 − 2 = 4.
- Say what Zara did. She added 3 + 4, which is the distance along the two dashed sides, not along the diagonal line AB. The diagonal is shorter.Zara found the length of the route along the two dashed lines.
- Use Pythagoras. AB is the hypotenuse.AB2 = 32 + 42 = 9 + 16 = 25
AB = √25 = 5
Check. 3, 4, 5 is a Pythagorean triple, and 5 is less than 7 as the hypotenuse must be less than the sum of the other two sides. ✓
Watch out. Going diagonally is shorter than going along the edges.
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 height using Pythagoras’ theorem on half of the triangle, then use area = ½ × base × height.
- Draw the height from A to the middle of BC. It splits the base into two halves of 8 cm and makes a right-angled triangle with hypotenuse 17 cm.Half the base = 16 ÷ 2 = 8 cm
- Find the height. We are finding a shorter side, so subtract the squares.h2 = 172 − 82 = 289 − 64 = 225
h = √225 = 15 cm - Find the area. Use the whole base 16 cm.Area = ½ × 16 × 15 = 120 cm2
- State the conclusion. The area is 120 cm2, as required.
Check. 8, 15, 17 is a Pythagorean triple: 64 + 225 = 289. ✓
Watch out. Use the whole base (16), not the half-base (8), in the area formula.
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 distance between two points is the hypotenuse of a right-angled triangle whose shorter sides are the horizontal and vertical steps between them.
- Draw a right-angled triangle between the two points. The line from one point to the other is the hypotenuse. The two shorter sides are the horizontal step and the vertical step.Horizontal step across, vertical step up or down.
- Find the horizontal and vertical steps. Subtract the x-coordinates, then subtract the y-coordinates. Only the size of each step matters, so ignore any minus sign.Horizontal: 5 − (−3) = 8, so the step is 8
Vertical: 5 − 1 = 4, so the step is 4 - Use Pythagoras’ theorem. The distance d is the hypotenuse.d2 = 82 + 42 = 64 + 16 = 80
- Take the square root and leave it exact. An exact answer is wanted, so we simplify the surd.d = √80
Find the biggest square number that divides 80: 80 = 16 × 5
√80 = √16 × √5 = 4√5
Check. 4√5 = 8.94427… on a calculator, and the line looks about that long on the grid. ✓
Watch out. Subtract the coordinates in the same order for x and y. A negative step is fine: squaring removes the minus sign.
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 the exact height of an equilateral triangle, then the area formula.
- Find the height with Pythagoras. The half-base is 5 cm and the hypotenuse is 10 cm.h2 = 102 − 52 = 100 − 25 = 75
- Simplify the surd. 75 = 25 × 3, and 25 is a square number.h = √75 = √25 × √3 = 5√3 cm
- Find the area. Area = ½ × base × height.Area = ½ × 10 × 5√3 = 25√3 cm2
Check. 25√3 = 43.30127…, and ½ × 10 × 8.66025… = 43.30127…. ✓
Watch out. Leave the answer as 25√3. Do not round when the question asks for an exact answer.
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. A regular hexagon splits into 6 equilateral triangles that meet at the centre. Find the area of one triangle, then multiply by 6.
- Find the height of one equilateral triangle. Each triangle has side 6 cm. The half-base is 3 cm.h2 = 62 − 32 = 36 − 9 = 27, so h = √27 = 3√3 cm
- Find the area of one triangle. ½ × base × height.½ × 6 × 3√3 = 9√3 cm2
- Multiply by 6. There are six identical triangles.6 × 9√3 = 54√3 cm2
- Give a decimal check. √3 = 1.7320…54 × 1.73205… = 93.53074… cm2 ≈ 93.5 cm2
Check. Another way: the hexagon area formula (3√3 ÷ 2) × side2 = 2.598… × 36 = 93.5. ✓
Watch out. A hexagon has 6 triangles, not 5 or 8.
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 the side be s. Use the right-angled triangle made by the height, half of the base and one side, then solve for s.
- Write Pythagoras’ theorem. The hypotenuse is s. The shorter sides are the height 10 and the half-base s ÷ 2.102 + (s ÷ 2)2 = s2
- Simplify. (s ÷ 2)2 = s2 ÷ 4.100 + s2÷4 = s2
- Collect the s terms. Subtract s2÷4 from both sides. s2 − s2÷4 = 3s2÷4.100 = 3s2÷4
- Solve for s2. Multiply both sides by 4, then divide by 3.400 = 3s2
s2 = 400 ÷ 3 = 133.33333… - Square root and round. We want s, not s2.s = √133.33333… = 11.54700…
s = 11.5 cm
Check. Put it back: half-base = 5.77; 5.772 + 102 = 133.3 and 11.552 = 133.3. ✓
Watch out. The height is a shorter side, so it is the half-base and the height that are squared and added to make the side squared.