Similarity - Teacher Slides.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Similarity

Similarity and congruence

Lesson 3 of 5

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up

Similarity and congruence

Lesson 3 of 5

Answer each one, then check.

1

What is the scale factor from 4 cm to 10 cm?

2

Write 3:5 as a fraction of the total.

3

Work out 9 × 1.5.

4

What does congruent mean?

5

What do the angles in a triangle add up to?

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up - Answers

Similarity and congruence

Lesson 3 of 5

1

What is the scale factor from 4 cm to 10 cm?

2.5

2

Write 3:5 as a fraction of the total.

⅜ and ⅝

3

Work out 9 × 1.5.

13.5

4

What does congruent mean?

Same shape and size

5

What do the angles in a triangle add up to?

180°

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

Similarity and congruence

Lesson 3 of 5

1

Explain what similar means.

2

Find scale factors between similar shapes.

3

Find missing lengths in similar shapes.

4

Show that two triangles are similar using equal angles.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Similarity and congruence

Lesson 3 of 5

SIMILAR

Two shapes are similar if they have the same shape but not necessarily the same size: their angles are equal and their lengths are in the same ratio.

The scale factor is the ratio of corresponding lengths.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Similar Triangles

Similarity and congruence

Lesson 3 of 5

Every side is multiplied by the same scale factor.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Facts About Similar Shapes

Similarity and congruence

Lesson 3 of 5

Angles

All corresponding angles are equal.

Lengths

All corresponding lengths are in the same ratio.

Scale factor

The number you multiply by to go from one shape to the other.

Congruent is a special case

A scale factor of 1.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Finding a Missing Length

Similarity and congruence

Lesson 3 of 5

Use the scale factor.

1

Match the corresponding sides

Find a pair where both lengths are known

2

Work out the scale factor

Larger ÷ smaller

3

Multiply or divide

Depending on the direction you go

4

Check

The answer should look sensible

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Missing Side

Similarity and congruence

Lesson 3 of 5

Triangles ABC and DEF are similar. AB = 6 cm, BC = 10 cm and AC = 8 cm. DE = 9 cm. Find EF and DF.

1

Scale factor

DE/AB = 9/6 = 1.5

2

EF corresponds to BC

10 × 1.5 = 15

3

DF corresponds to AC

8 × 1.5 = 12

ANSWER

EF = 15 cm and DF = 12 cm.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Similar Rectangles

Similarity and congruence

Lesson 3 of 5

Rectangle A is 4 cm by 6 cm. Rectangle B is similar to A and 10 cm long on its shorter side. Find the other side of B.

1

Scale factor

10 ÷ 4 = 2.5

2

Other side

6 × 2.5

ANSWER

15 cm

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Showing Triangles Are Similar

Similarity and congruence

Lesson 3 of 5

In triangle PQR, angle P = 50° and angle Q = 60°. In triangle XYZ, angle X = 50° and angle Z = 70°. Show that the triangles are similar.

1

Find angle R

180 − 50 − 60 = 70°

2

Find angle Y

180 − 50 − 70 = 60°

3

All three angles match

50°, 60°, 70°

ANSWER

The triangles have equal angles, so they are similar.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Similar or Congruent?

Similarity and congruence

Lesson 3 of 5

SIMILAR

• Same shape, possibly different size.

• Equal angles, sides in the same ratio.

• Scale factor may be any number.

CONGRUENT

• Same shape and same size.

• Equal angles and equal sides.

• Scale factor 1.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Key Terms

Similarity and congruence

Lesson 3 of 5

Similar

The same shape but a different size.

Scale factor

The multiplier that takes one length to the corresponding length.

Corresponding sides

Sides in matching positions.

Ratio

A comparison of two quantities.

Enlargement

A transformation that produces similar shapes.

Equiangular

Having equal angles.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Similarity and congruence

Lesson 3 of 5

YOUR TASK

Photo Enlargement

10 minutes

A photograph is 12 cm by 8 cm. It is enlarged to fit a frame that is 30 cm wide. Find the height of the enlargement and the scale factor. Then say whether a 30 cm by 18 cm frame would suit the photograph.

1

Find the scale factor.

2

Apply it to the other side.

3

Compare ratios.

WHAT A GOOD ANSWER SHOWS

Scale factor 30 ÷ 12 = 2.5, so the height is 8 × 2.5 = 20 cm. A 30 by 18 frame is not similar to 12 by 8, because 18 ÷ 8 = 2.25 ≠ 2.5, so the picture would be distorted.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Can I...?

Similarity and congruence

Lesson 3 of 5

Say what similar means.

Find a scale factor.

Find a missing length.

Use scale factors greater than 1.

Use scale factors less than 1.

Show triangles are similar by angles.

Identify corresponding sides.

Explain the difference from congruent.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Summary & Exam Focus

Similar shapes have equal angles and proportional sides.

Scale factor = new length ÷ old length.

Match corresponding sides carefully.

Two equal angles are enough to show triangles are similar.

EXAM FOCUS

Triangles ABC and DEF are similar. AB = 6 cm, BC = 10 cm, DE = 9 cm. Work out the length of EF. (3 marks)

Find the scale factor from a pair of sides you know, then use it on the side you need. Make sure the sides really correspond.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Exam Practice: Similarity

Similarity and congruence

Lesson 3 of 5

Answer all questions. Show your working. · 25 minutes

Question 1 · 3 marks · Work out

The diagram shows two similar triangles, ABC and DEF. AB = 6 cm, BC = 10 cm, AC = 8 cm and DE = 9 cm. Work out the length of EF.

Question 2 · 2 marks · Work out

For the triangles in the last question, work out the length of DF.

Question 3 · 3 marks · Work out

Rectangle A is 4 cm by 6 cm. Rectangle B is similar to rectangle A. The shorter side of rectangle B is 10 cm. Work out the length of the…

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Exam Practice: Similarity (cont.)

Similarity and congruence

Lesson 3 of 5

Question 4 · 3 marks · Show that

In triangle PQR, angle P = 50° and angle Q = 60°. In triangle XYZ, angle X = 50° and angle Z = 70°. Show that triangles PQR and XYZ are…

Question 5 · 2 marks · Explain

A photograph is 12 cm by 8 cm. It is enlarged so that its width is 30 cm. Work out the height of the enlarged photograph.

Question 6 · 2 marks · Decide

Are a 3 cm by 5 cm rectangle and a 9 cm by 14 cm rectangle similar? Give a reason.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 1 · 3 marks · Work out

Similarity and congruence

Lesson 3 of 5

The diagram shows two similar triangles, ABC and DEF. AB = 6 cm, BC = 10 cm, AC = 8 cm and DE = 9 cm. Work out the length of EF. (3 marks)

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 1 · mark scheme

Similarity and congruence

Lesson 3 of 5

3 marks available. Award a mark for each point made.

Scale factor 1.5

M1

10 × 1.5

M1

15

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 2 · 2 marks · Work out

Similarity and congruence

Lesson 3 of 5

“

For the triangles in the last question, work out the length of DF.

HOW TO ANSWER IT

Command word: Work out. Worth 2 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 2 · mark scheme

Similarity and congruence

Lesson 3 of 5

2 marks available. Award a mark for each point made.

8 × 1.5

M1

12

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 3 · 3 marks · Work out

Similarity and congruence

Lesson 3 of 5

“

Rectangle A is 4 cm by 6 cm. Rectangle B is similar to rectangle A. The shorter side of rectangle B is 10 cm. Work out the length of the longer side of rectangle B.

HOW TO ANSWER IT

Command word: Work out. Worth 3 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 3 · mark scheme

Similarity and congruence

Lesson 3 of 5

3 marks available. Award a mark for each point made.

Scale factor 2.5

M1

6 × 2.5

M1

15

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 4 · 3 marks · Show that

Similarity and congruence

Lesson 3 of 5

“

In triangle PQR, angle P = 50° and angle Q = 60°. In triangle XYZ, angle X = 50° and angle Z = 70°. Show that triangles PQR and XYZ are similar.

HOW TO ANSWER IT

Command word: Show that. Worth 3 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 4 · mark scheme

Similarity and congruence

Lesson 3 of 5

3 marks available. Award a mark for each point made.

Finding a third angle

M1

Finding the other third angle

M1

All angles equal, so similar

C1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 5 · 2 marks · Explain

Similarity and congruence

Lesson 3 of 5

“

A photograph is 12 cm by 8 cm. It is enlarged so that its width is 30 cm. Work out the height of the enlarged photograph.

HOW TO ANSWER IT

Command word: Explain. Worth 2 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 5 · mark scheme

Similarity and congruence

Lesson 3 of 5

2 marks available. Award a mark for each point made.

30 ÷ 12

M1

20

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 6 · 2 marks · Decide

Similarity and congruence

Lesson 3 of 5

“

Are a 3 cm by 5 cm rectangle and a 9 cm by 14 cm rectangle similar? Give a reason.

HOW TO ANSWER IT

Command word: Decide. Worth 2 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 6 · mark scheme

Similarity and congruence

Lesson 3 of 5

2 marks available. Award a mark for each point made.

Comparing the ratios

M1

Not similar with a reason

C1

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