1MA1 · G5–G6, G19Geometry & MeasuresFoundation & HigherGrades 4–8

Congruence & Similarity

Q: Why does area scale by k²?

Area multiplies two lengths, and each is multiplied by k, giving k k.

Two shapes are $\textbf{congruent}$ if they are exactly the same shape and size. They are $\textbf{similar}$ if one is an enlargement of the other — same shape, possibly different size. Similarity is surprisingly powerful: it finds lengths you cannot measure, and at Higher tier it tells you how areas and volumes change when a shape is scaled.

50 min Transformations & Vectors

The big picture

For triangles, you do not need to check every side and angle to prove congruence: four sets of conditions — SSS, SAS, ASA and RHS — are each enough. Writing a congruence proof means matching three pieces of information and naming the condition.

In similar shapes, every length is multiplied by the same scale factor $k$ . But areas are two-dimensional, so they scale by $k^{2}$ , and volumes by $k^{3}$ . That jump from $k$ to $k^{2}$ to $k^{3}$ is the core Higher-tier idea, and a frequent source of lost marks.

What you'll learn

Use SSS, SAS, ASA and RHS to prove triangles congruent
Find missing lengths in similar shapes using a scale factor
Show that two triangles are similar
Use area and volume scale factors for similar shapes (Higher)

Congruent triangles

Two triangles are congruent if any one of these holds: $\textbf{SSS}$ — all three sides equal; $\textbf{SAS}$ — two sides and the included angle equal; $\textbf{ASA}$ — two angles and a corresponding side equal; $\textbf{RHS}$ — right angle, hypotenuse and one other side equal.

AAA is not enough: triangles with the same angles can be different sizes. SSA is not generally enough either.

In a proof, list the three matching facts with reasons, then state the condition.

Example 1

ABCD

is a parallelogram. Prove that triangles

ABD

and

CDB

are congruent.

AB = CD

: opposite sides of a parallelogram are equal.

AD = CB

: opposite sides of a parallelogram are equal.

BD

is common to both triangles.

AnswerSo triangles

ABD

and

CDB

are congruent (SSS).

Tip — Match vertices in order: in " $ABD$ and $CDB$ ", $A$ corresponds to $C$ , $B$ to $D$ , and $D$ to $B$ . Getting the order right shows which sides match.

Similar shapes and scale factors

In similar shapes, corresponding angles are equal and corresponding lengths are in the same ratio. The linear scale factor is $k = \dfrac{\text{length on larger}}{\text{corresponding length on smaller}}$ .

Multiply by $k$ to go from small to large; divide by $k$ to go back.

Two triangles are similar if their angles are equal (AA is enough, since the third angle follows), or if all three pairs of sides are in the same ratio.

Example 1Triangles

PQR

and

XYZ

are similar.

PQ = 6

cm,

QR = 9

cm,

XY = 10

cm. Find

YZ

PQ

corresponds to

XY

k = \dfrac{10}{6} = \dfrac53

YZ = 9 \times \dfrac53

Answer

YZ = 15

Your turnSame method, new numbers — try it before moving on.

Two similar triangles have corresponding sides 8 cm and 12 cm. Another side of the smaller triangle is 10 cm. Find the matching side of the larger, in cm.

Your answer

Show worked solution

Solution

1Scale factor $= 12 \div 8 = 1.5$ .
2 $10 \times 1.5 = 15$ cm.

Answer: $15$ cm

Why this works

In a diagram where one triangle sits inside another — a line drawn parallel to one side — the two triangles are similar because parallel lines give equal corresponding angles. Redraw them separately to match sides confidently.

Area and volume scale factors

If lengths are multiplied by $k$ , areas are multiplied by $k^{2}$ and volumes by $k^{3}$ .

So a model built at scale factor 3 has 9 times the surface area and 27 times the volume.

Working backwards, take the square root of an area ratio or the cube root of a volume ratio to find $k$ .

\text{length} \times k,\ \text{area} \times k^{2},\ \text{volume} \times k^{3}

Similar shapes.

Example 1Two similar cylinders have heights 6 cm and 9 cm. The smaller has volume 400 cm³. Find the volume of the larger.

k = \dfrac{9}{6} = 1.5

2Volume factor

k^{3} = 3.375

400 \times 3.375

Answer1350 cm³

Your turnSame method, new numbers — try it before moving on.

Two similar solids have lengths in the ratio $2 : 5$ . The smaller has volume 40 cm³. Find the volume of the larger, in cm³.

Your answer

Show worked solution

Solution

1Volume scale factor $= \left(\frac{5}{2}\right)^3 = \frac{125}{8}$ .
2 $40 \times \frac{125}{8} = 625$ cm³.

Answer: $625$ cm³

Example 2Two similar shapes have areas 20 cm² and 45 cm². The smaller has a side of 4 cm. Find the corresponding side of the larger.

1Area factor:

\dfrac{45}{20} = 2.25

2Length factor:

\sqrt{2.25} = 1.5

4 \times 1.5

Answer6 cm

Your turnSame method, new numbers — try it before moving on.

Two similar shapes have areas 18 cm² and 50 cm². A side of the smaller is 6 cm. Find the matching side of the larger, in cm.

Your answer

Show worked solution

Solution

1Area factor $\frac{50}{18} = \frac{25}{9}$ , so length factor $\frac{5}{3}$ .
2 $6 \times \frac{5}{3} = 10$ cm.

Answer: $10$ cm

Tip — Always find the linear scale factor $k$ first, whichever quantity you are given. Then square or cube it as needed.

Think like an examiner

In congruence proofs, give a reason for each matching pair and name the condition.
Match corresponding sides carefully before finding a scale factor.
Show the square or cube root when working back from area or volume.

Scale factors

k = \tfrac{\text{new length}}{\text{old length}}

Linear scale factor.

k^{2}

Area scale factor.

k^{3}

Volume scale factor.

Watch out for these

Using AAA to prove congruence.

Equal angles show similarity, not congruence.

Multiplying an area by the linear scale factor.

Areas scale by

k^{2}

Dividing a volume ratio by 3 to get

k

Take the cube root.

Pairing sides by position on the page.

Pair sides opposite equal angles.

Stretch yourself

Two similar bottles have surface areas of 180 cm² and 405 cm². The larger holds 1.35 litres. How much does the smaller hold?

Hint — Area factor → linear factor → volume factor.

Reveal solution

SolutionArea factor

= \dfrac{405}{180} = 2.25

, so the linear factor is

k = 1.5

and the volume factor is

1.5^{3} = 3.375

. The smaller bottle holds

1.35 \div 3.375 = 0.4

litres, or 400 ml.

\blacksquare

Questions students ask

Are all squares similar?

Yes. All squares have equal angles and sides in the same ratio. The same is true of all circles and all equilateral triangles.

Why does area scale by k²?

Area multiplies two lengths, and each is multiplied by

k

, giving

k \times k

What is the difference between ASA and AAS?

Both work: once two angles are equal the third is too, so any corresponding side fixes the size.

Key takeaways

Congruent: same shape and size. Conditions SSS, SAS, ASA, RHS.
Similar: same shape; lengths in ratio $k$ ; equal angles.
Find $k$ from a pair of corresponding lengths.
Area scales by $k^{2}$ , volume by $k^{3}$ .
Work back with square or cube roots.

Exam-style questions

Practise it the way it is examined

3 original questions · 8 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 53 marks

In triangle $ABC$ , $AB = AC$ . $M$ is the midpoint of $BC$ .

Prove that triangles $ABM$ and $ACM$ are congruent.[3]
Show mark scheme
- B1 $AB = AC$ (given) and $BM = CM$ ( $M$ is the midpoint)
- B1 $AM$ is a common side
- C1Congruent by SSS

Question 2Grade 62 marks

In triangle $ADE$ , $B$ is on $AD$ and $C$ is on $AE$ with $BC$ parallel to $DE$ . $AB = 8$ cm, $AD = 12$ cm and $BC = 6$ cm.

Work out the length of $DE$ .[2]
Show mark scheme
- M1Scale factor $\frac{12}{8} = 1.5$
- A1 $9$ cm
Answer: $9$ cm

Question 3Grade 83 marks

Two similar bottles have heights 12 cm and 18 cm. The larger bottle holds 1080 ml. How much does the smaller bottle hold?[3]
Show mark scheme
- M1Length factor $\frac{2}{3}$
- M1Volume factor $\frac{8}{27}$
- A1 $320$ ml
Answer: $320$ ml

Unlimited practice

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