EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Pythagoras' theorem 2
Angles and trigonometry · Lesson 5 of 7
Last Lesson and Before
Answer each one, then check.
1. Last lesson: a right-angled triangle has shorter sides 6 and 8. How long is the hypotenuse?
10
2. What is the formula for the area of a triangle?
½ × base × height
3. Work out 4 − (−2).
6
4. Is 5, 12, 13 a Pythagorean triple?
Yes: 25 + 144 = 169
Learning Objectives
1. Find the distance between two points on a coordinate grid.
2. Find the height of an isosceles triangle, and use it to find the area.
3. Spot the right-angled triangle hidden inside a shape.
4. Solve problems that need Pythagoras' theorem twice.
Finding the Right-Angled Triangle
Most problems don't hand you a right-angled triangle - you have to draw it in.
▸ Isosceles triangle. The line of symmetry cuts it into two right-angled triangles, and halves the base.
▸ Rectangle. A diagonal makes two right-angled triangles.
▸ Trapezium. Drop a vertical height from a top corner to make a right-angled triangle at the end.
▸ Coordinates. The horizontal and vertical distances between two points are the two shorter sides.
The Distance Between Two Points
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Subtract the x-coordinates to get the horizontal side and the y-coordinates to get the vertical side. Then AB = √(9² + 5²) = √106 = 10.3 to 1 decimal place. |
The line joining two points is the hypotenuse of a right-angled triangle. |
The Distance Between Two Points
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Work out the length of the line joining A(1, 2) and B(10, 7). Give your answer to 1 decimal place. |
1. Horizontal distance
10 − 1 = 9
2. Vertical distance
7 − 2 = 5
3. Pythagoras
AB² = 9² + 5² = 81 + 25 = 106
4. Square root
AB = √106 = 10.295…
Answer: 10.3 units
Area of an Isosceles Triangle
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An isosceles triangle has sides 10 cm, 10 cm and 12 cm. Work out its area. |
1. The line of symmetry halves the base
Half base = 6 cm, hypotenuse 10 cm
2. Pythagoras for the height
h² = 10² − 6² = 100 − 36 = 64, so h = 8
3. Area
½ × 12 × 8 = 48
Answer: 48 cm²
The Slant Side of a Trapezium
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An isosceles trapezium has parallel sides 10 cm and 16 cm, and a height of 8 cm. Work out its perimeter, to 1 decimal place. |
1. The extra length on the longer side is shared between the two ends
(16 − 10) ÷ 2 = 3 cm
2. Pythagoras for one slant side
s² = 3² + 8² = 73, so s = 8.544…
3. Add the four sides
10 + 16 + 2 × 8.544…= 43.088…
Answer: 43.1 cm
Two Triangles, Two Steps
When two right-angled triangles share a side, find the shared side first.
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1 Sketch Draw both triangles and mark the right angles. |
2 Shared side Find the side the triangles share, using the triangle where you know two sides. |
3 Keep it exact Keep the full value (or its square) in your calculator - don't round yet. |
4 Second triangle Use the shared side to find the length asked for. |
5 Round last Round only the final answer. |
Key Terms
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Hypotenuse The longest side of a right-angled triangle, opposite the right angle. |
Line of symmetry A line that splits a shape into two mirror-image halves. |
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Perpendicular height The height measured at right angles to the base. |
Coordinates A pair of numbers (x, y) giving the position of a point. |
Your Task: Shortest Route
12 minutes
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A spider is in one corner of the floor of a room 4 m long and 3 m wide. A fly is in the opposite corner of the floor. (a) How far does the spider walk if it goes along two walls? (b) How far if it walks straight across the floor? (c) Points P(−3, 1) and Q(5, 7) are two towns on a map grid in km. How far apart are they? 1. Sketch the right-angled triangle. 2. Label the two shorter sides. 3. Use Pythagoras. |
A good answer shows: (a) 4 + 3 = 7 m (b) √(16 + 9) = 5 m (c) Horizontal 8, vertical 6, so √(64 + 36) = 10 km.
Can I...?
☐ Find the distance between two points on a grid.
☐ Find the height of an isosceles triangle.
☐ Find the area of an isosceles triangle.
☐ Find the slant side of a trapezium.
☐ Use Pythagoras twice with two triangles.
Summary
✓ Distance between points: horizontal and vertical differences are the shorter sides.
✓ Isosceles triangle: the line of symmetry halves the base and makes a right angle.
✓ Two triangles: find the shared side first.
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EXAM FOCUS Triangle ABC is right-angled at B, with AB = 5 cm and BC = 12 cm. Triangle ACD is right-angled at D, with CD = 5 cm. Work out the length of AD. (4 marks) Mark every right angle on the diagram before you start. Each Pythagoras step uses one right-angled triangle - write which one. |