Pythagoras theorem 2.pptx

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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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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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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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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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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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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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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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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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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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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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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