Good Learning Co.
SavannahYear 8
← Maths
MathsShape, probability and data30 min★★★ difficulty

Congruence, similarity and enlargement

Congruent shapes are identical; similar shapes are the same shape at a different size. Distinguishing the two, and proving which you have, is the heart of geometric reasoning.

Part of your national curriculum
  • Geometry and measures: Identify and construct congruent triangles, and construct similar shapes by enlargement, with and without coordinate grids

Lesson overview

What you'll learn in this lesson

Identify and construct congruent triangles, and construct similar shapes by enlargement, with and without coordinate grids

Key learning points

  • • Conditions for congruence
  • • Enlargement from a centre
  • • What enlargement preserves
  • • Using similarity to find lengths

This lesson at a glance

  • 30 minutes
  • 17 parts to scroll through
  • 4 quick checks
  • Marked quiz at the end
  • Gentle pace: short sittings with pauses

Words to know

constructcongruenttrianglessimilarshapes

Scroll down — the lesson carries on below

Part 1 of 17 · Discover6%
1

Watch & discover

Part 1 of 17

Watch first

Congruence, similarity and enlargement

An English teaching video on this topic from mathantics. Watch it, then scroll down for the lesson.

In a nutshell

Identify and construct congruent triangles, and construct similar shapes by enlargement, with and without coordinate grids

2

Learning cycle

Part 2 of 17

Learning cycle 1 of 2

Part 1 · Conditions for congruence

A short piece of teaching, then a check to make sure it has landed.

3

Explore the idea

Part 3 of 17

Learn

Conditions for congruence

Two triangles are congruent if they match on SSS, SAS, ASA or RHS. Note that SSA is not a condition: two sides and a non-included angle can produce two different triangles. Naming the condition you are using is required in a congruence proof.

4

Explore the idea

Part 4 of 17

Learn

Enlargement from a centre

An enlargement needs a scale factor and a centre. From centre (0, 0) with factor 2, the point (3, 1) maps to (6, 2): each coordinate distance from the centre doubles. A fractional scale factor such as 1/2 produces a smaller image, and a negative factor puts the image on the opposite side of the centre.

5

Quick check

Part 5 of 17

Quick check

Which is NOT a congruence condition?

6

Quick check

Part 6 of 17

Quick check

Enlargement scale factor 1/2 produces an image that is

7

Learning cycle

Part 7 of 17

Learning cycle 2 of 2

Part 2 · What enlargement preserves

A short piece of teaching, then a check to make sure it has landed.

8

Reset break

Part 8 of 17

Pause

That's sitting 1 of 2 done

Stretch, get a drink, look out of the window. There is no timer and nothing is counting down — your place is saved, so you can come back in five minutes or tomorrow.

Stop here for now
9

Explore the idea

Part 9 of 17

Learn

What enlargement preserves

Enlargement preserves angles and the ratios of sides, so the image is similar to the object. Lengths change by k, areas by k² and volumes by k³. Describing a transformation fully means stating that it is an enlargement, giving the scale factor and giving the centre.

10

Explore the idea

Part 10 of 17

Learn

Using similarity to find lengths

In similar triangles, match corresponding vertices before writing any ratio. Draw both triangles the same way up, label matching sides, then form the scale factor from the pair where both lengths are known.

11

Quick check

Part 11 of 17

Quick check

Enlarging by factor 4 multiplies volume by

12

Quick check

Part 12 of 17

Quick check

A full description of an enlargement needs

13

Explore the idea

Part 13 of 17

Worked example

Worked answer: describe fully the transformation mapping (3, 1) to (6, 2) and (4, 3) to (8, 6) (3 marks)

Each coordinate has doubled, so the scale factor is 2 (1). Rays through each object point and its image meet at the origin, so the centre of enlargement is (0, 0) (1). The full description is: an enlargement, scale factor 2, centre (0, 0) (1). Naming only 'enlargement' without both the factor and the centre would score one mark at most.

14

Challenge round

Part 14 of 17

Game · Sort it

Which of these are true?

Drag each card into the right column. Tap a card first if dragging is fiddly.

True

Not true

15

Challenge round

Part 15 of 17

Game · Fill the gaps

Finish the sentences

Choose the word that belongs in each gap.

Two triangles are ____ if they match on SSS, SAS, ASA or RHS.

Note that SSA is not a condition: two sides and a non-included angle can produce two different ____.

Enlargement preserves angles and the ratios of sides, so the image is ____ to the object.

16

Mastery quiz

Part 16 of 17

Marked quiz

End of lesson quiz: Congruence, similarity and enlargement

4 questions, marked with the reasoning shown. No timer.

  1. 1. Which is NOT a congruence condition?

  2. 2. Enlargement scale factor 1/2 produces an image that is

  3. 3. Enlarging by factor 4 multiplies volume by

  4. 4. A full description of an enlargement needs

17

Lesson round-up

Part 17 of 17

Lesson round-up

Ready when you are

Quiz score

Not sat

Games

Not played

Points this lesson

0

Best run

0 in a row

Luna: 0 out of 4 on the practice checks. Only if you feel up to it — one more?

Ask Luna

Do the work

Your written work for this lesson

This is the part you'd do in your book at school. Pick the pieces below, write here or on paper, and photograph anything you draw or work out.

Article

The cost of a school trip

A school is planning a trip to a wildlife park. The coach costs £480 however many people go. Entry costs £9.50 for each pupil and each adult goes free. The school says every pupil must pay the same amount, and the trip must not lose money.

If 60 pupils go, the coach works out at £8 each, so the total for each pupil is £17.50. If only 40 pupils go, the coach works out at £12 each, so the price rises to £21.50. The teacher explains that the coach is a fixed cost and the entry is a cost per person — which is why the price per pupil changes when the number changes.

The school decides to charge £20 and to use anything left over for a second trip later in the year.

Written for Good Learning Co.

Reading questions

  1. Work out the total cost of the trip if 50 pupils go.
  2. Show why the price per pupil falls when more pupils go, using numbers.
  3. At £20 each, how many pupils must go before the trip makes money? Show your working.

  • Working out

    Working out

    Do the practice questions on paper, showing every step — not just the answer. Mark them yourself and circle anything you got wrong.

    10–15 questions

  • Worked example, explained

    Writing

    Copy one question out and write underneath, in words, why each step works.

    1 question, 4–5 sentences

  • Problem in words

    Working out

    Write your own real-life problem that uses today's method, then solve it.

    1 problem + solution

Saved as you type.

Part 1 of 17 · Watch & discover

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    Identify and construct congruent triangles, and construct similar shapes by enlargement, with and without coordinate grids

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