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Pythagoras theorem 2.pptx
Built from the lesson script on 29 September 2026.
EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Pythagoras' theorem 2
Angles and trigonometry
Lesson 5 of 7
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Last Lesson and Before
Angles and trigonometry
Lesson 5 of 7
Answer each one, then check.
1
Last lesson: a right-angled triangle has shorter sides 6 and 8. How long is the hypotenuse?
2
What is the formula for the area of a triangle?
3
Work out 4 − (−2).
4
Is 5, 12, 13 a Pythagorean triple?
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Last Lesson and Before - Answers
Angles and trigonometry
Lesson 5 of 7
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
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Learning Objectives
Angles and trigonometry
Lesson 5 of 7
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.
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Finding the Right-Angled Triangle
Angles and trigonometry
Lesson 5 of 7
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.
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
The Distance Between Two Points
Angles and trigonometry
Lesson 5 of 7

The line joining two points is the hypotenuse of a right-angled triangle.
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.
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
The Distance Between Two Points
Angles and trigonometry
Lesson 5 of 7
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
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Area of an Isosceles Triangle
Angles and trigonometry
Lesson 5 of 7
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
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
The Slant Side of a Trapezium
Angles and trigonometry
Lesson 5 of 7
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
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Two Triangles, Two Steps
Angles and trigonometry
Lesson 5 of 7
When two right-angled triangles share a side, find the shared side first.
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.
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Key Terms
Angles and trigonometry
Lesson 5 of 7
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.
Perpendicular height
The height measured at right angles to the base.
Coordinates
A pair of numbers (x, y) giving the position of a point.
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Angles and trigonometry
Lesson 5 of 7
YOUR TASK
Shortest Route
12 minutes
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.
WHAT A GOOD ANSWER SHOWS
(a) 4 + 3 = 7 m (b) √(16 + 9) = 5 m (c) Horizontal 8, vertical 6, so √(64 + 36) = 10 km.
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Can I...?
Angles and trigonometry
Lesson 5 of 7
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.
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Summary & Exam Focus
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.
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.
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