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AQA MathsGeometry and measures

Congruence and similarity

Use matching shapes and scale factors.

AQAGCSE MathsGeometry and measuresFoundation and Higher
Visual model

Similar shapes keep angles but scale lengths

shape Asame angles, bigger lengthsscale factor
Gold-standard guide
20 mins

What you will learn

Use matching shapes and scale factors.
Use a clear step-by-step method for congruence and similarity.
Check your answer and avoid the most common exam mistake.
Useful before you start
Core number skillsEarlier geometry and measures skillsShowing clear working
Core knowledge

Know the rule, then use it

These are the short notes. Read each one, then check you can use it in the worked example below.

Method

Method

Congruent shapes: identical shape and size — corresponding sides and angles all equal

Step 1

Find the linear scale factor

k = 96\frac{9}{6} = 32\frac{3}{2} = 1.5

Step 2

Find the area scale factor

Area SF = k2 = 1.52 = 2.25

Step 3

Multiply to find the larger area

Area = 24 × 2.25 = 54 cm2

Watch out

Watch out

Students use the linear scale factor to scale areas (multiplying by 1.5 instead of 2.25)

f
Similar lengths

newlength=oldlength×scalefactor.new length = old length \times scale factor.

f
Similar solids

areascalefactor=k2andvolumescalefactor=k3.area scale factor = k^{2} and volume scale factor = k^{3}.

Worked example

Two similar triangles have corresponding sides of 6 cm and 9 cm. The area of the smaller triangle is 24 cm2. Find the area of the larger triangle.

1

Find the linear scale factor: k = 96\frac{9}{6} = 32\frac{3}{2} = 1.5.

2

Find the area scale factor: Area SF = k2 = 1.52 = 2.25.

3

Multiply to find the larger area: Area = 24 × 2.25 = 54 cm2.

Final answer

54 cm2

Question ladder

Build up to the hardest questions

Do them in order. If you miss a step, read the solution, then redo the question without looking.

Workedreasoning

Two similar triangles have corresponding sides of 6 cm and 9 cm. The area of the smaller triangle is 24 cm2. Find the area of the larger triangle.

3 marks4 minscongruence-and-similarity-worked
Show solution
Worked solution
  1. 1.Find the linear scale factor: k = 96\frac{9}{6} = 32\frac{3}{2} = 1.5.
  2. 2.Find the area scale factor: Area SF = k2 = 1.52 = 2.25.
  3. 3.Multiply to find the larger area: Area = 24 × 2.25 = 54 cm2.
Final answer

54 cm2

Mark points
  • M1: find the linear scale factor
  • M1: find the area scale factor
  • M1: multiply to find the larger area
  • A1: 54 cm2
Watch out

25). Always square the linear SF for area, cube it for volume. 25' as a clear line prevents this error.

Diagnosticrecall

Two similar solids have linear SF 2:5. Find the volume SF.

1 mark2 minscongruence-and-similarity-q1
Show solution
Worked solution
  1. 1.Spot the skill: Congruent shapes: identical shape and size — corresponding sides and angles all equal.
  2. 2.Use the find the linear scale factor stage first, then find the area scale factor.
  3. 3.Keep the final answer visible: 8:125.
Final answer

8:125

Mark points
  • M1: use the correct congruent shapes: identical shape and size — corresponding sides and angles all equal.similar shapes: same shape, different size.linear scale factor k means area scale factor k2 and volume scale factor k3.
  • A1: 8:125
Watch out

25). Always square the linear SF for area, cube it for volume. 25' as a clear line prevents this error.

Easyprocedure

A photo is enlarged from 4 cm × 6 cm to 10 cm × 15 cm. Find the linear SF.

2 marks3 minscongruence-and-similarity-q2
Show solution
Worked solution
  1. 1.Spot the skill: Congruent shapes: identical shape and size — corresponding sides and angles all equal.
  2. 2.Use the find the area scale factor stage first, then multiply to find the larger area.
  3. 3.Keep the final answer visible: 2.5.
Final answer

2.5

Mark points
  • M1: use the correct congruent shapes: identical shape and size — corresponding sides and angles all equal.similar shapes: same shape, different size.linear scale factor k means area scale factor k2 and volume scale factor k3.
  • A1: 2.5
Watch out

25). Always square the linear SF for area, cube it for volume. 25' as a clear line prevents this error.

Mediumreasoning

Two similar rectangles have areas 36 cm2 and 81 cm2. Find the ratio of their perimeters.

3 marks4 minscongruence-and-similarity-q3
Show solution
Worked solution
  1. 1.Spot the skill: Congruent shapes: identical shape and size — corresponding sides and angles all equal.
  2. 2.Use the multiply to find the larger area stage first, then find the linear scale factor.
  3. 3.Keep the final answer visible: 2:3.
Final answer

2:3

Mark points
  • M1: use the correct congruent shapes: identical shape and size — corresponding sides and angles all equal.similar shapes: same shape, different size.linear scale factor k means area scale factor k2 and volume scale factor k3.
  • A1: 2:3
Watch out

25). Always square the linear SF for area, cube it for volume. 25' as a clear line prevents this error.

Hardproblem solving

State the condition that makes two triangles congruent given two sides and the included angle.

3 marks5 minscongruence-and-similarity-q4
Show solution
Worked solution
  1. 1.Spot the skill: Congruent shapes: identical shape and size — corresponding sides and angles all equal.
  2. 2.Use the find the linear scale factor stage first, then find the area scale factor.
  3. 3.Keep the final answer visible: SAS (side-angle-side).
Final answer

SAS (side-angle-side)

Mark points
  • M1: use the correct congruent shapes: identical shape and size — corresponding sides and angles all equal.similar shapes: same shape, different size.linear scale factor k means area scale factor k2 and volume scale factor k3.
  • A1: SAS (side-angle-side)
Watch out

25). Always square the linear SF for area, cube it for volume. 25' as a clear line prevents this error.

Exam-stylemulti-step

Two similar prisms have volumes 64 cm3 and 216 cm3. Find the linear SF.

4 marks6 minscongruence-and-similarity-q5
Show solution
Worked solution
  1. 1.Spot the skill: Congruent shapes: identical shape and size — corresponding sides and angles all equal.
  2. 2.Use the find the area scale factor stage first, then multiply to find the larger area.
  3. 3.Keep the final answer visible: 2:3.
Final answer

2:3

Mark points
  • M1: use the correct congruent shapes: identical shape and size — corresponding sides and angles all equal.similar shapes: same shape, different size.linear scale factor k means area scale factor k2 and volume scale factor k3.
  • A1: 2:3
Watch out

25). Always square the linear SF for area, cube it for volume. 25' as a clear line prevents this error.

Grade 9 stretchproblem solving

Two similar solids have length scale factor 3:5. The smaller volume is 216 cm3. Find the larger volume.

3355k3k^{3}
4 marks7 minssimilarity-g9
Show solution
Worked solution
  1. 1.Cube the length scale factor.
  2. 2.Multiply by the volume scale factor 1252712\frac{5}{2}7.
Final answer

1000 cm3

Mark points
  • M1: volume ratio 27:125
  • M1: 216 × 1252712\frac{5}{2}7
  • A1: 1000 cm3
Watch out

Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.

Timed checkpoint
12 mins - 9 marks

Switch between skills

Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.

1Congruence and similarity - 2 marksTwo similar solids have linear SF 2:5. Find the volume SF.Mark answer
Answer

8:125

2Angles, lines and polygons - 2 marksThe exterior angle of a regular polygon is 24°. How many sides?Mark answer
Answer

15

3Properties of shapes - 2 marksName all 2D shapes with equal diagonals that bisect each other.Mark answer
Answer

Rectangle, square

4Perimeter, area and volume - 3 marksFind the perimeter of a rectangle with length 13 cm and width 8 cm.Mark answer
Answer

42 cm

Mastery check
  • I can explain the method for congruence and similarity.
  • I can show clear working without skipping key steps.
  • 25). Always square the linear SF for area, cube it for volume. 25' as a clear line prevents this error.
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Related topics
Angles, lines and polygonsProperties of shapesPerimeter, area and volume
Official exam-board sources

This guide follows the AQA GCSE Mathematics 8300 specification. Practice questions are original Learnova questions shaped around official content and exam skills.

Official specification Official past papers
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