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Core notes

Congruent figures match in size and shape; similar figures match in shape with a common scale factor. Establish the correspondence before comparing sides, areas or volumes.

Congruence and similarity

Congruent figures have equal corresponding lengths and angles. Similar figures have equal corresponding angles and proportional corresponding sides. Determine the correct vertex correspondence before forming any ratio.

Enlargement, reduction and scale drawings

An enlargement or reduction multiplies every length from the centre by one scale factor. A scale drawing represents actual dimensions proportionally, so convert units before applying the scale.

Bisectors and constructions

A perpendicular bisector contains points equidistant from a segment's endpoints. An angle bisector contains points equidistant from the angle's two sides. Construct each locus with intersecting compass arcs.

Area and volume scale factors

If corresponding lengths scale by $k$ , areas scale by $k^2$ and volumes by $k^3$ . Take the appropriate root when working backwards from an area or volume ratio.

Relationship

Use

\text{area ratio}=k^2

Convert a linear scale factor into an area ratio.

\text{volume ratio}=k^3

Convert a linear scale factor into a volume ratio.

Worked examples

Example 1: Two similar solids have corresponding lengths in ratio $2:5$ . Find their volume ratio.

The linear scale factor is $\frac{2}{5}$ .
Cube both terms for volume.
Calculate $2^3:5^3=8:125$ .

Answer: The volume ratio is $8:125$ .

Example 2: A model uses scale $1:200$ . A wall measures $7.5\text{ cm}$ on the model. Find its actual length in metres.

Multiply by $200$ : $7.5\times200=1500\text{ cm}$ .
Convert centimetres to metres.
Calculate $1500\div100=15\text{ m}$ .

Answer: The actual wall is $15\text{ m}$ long.

Exam traps and retrieval check

Avoid these traps

Pairing non-corresponding sides.
Using the linear scale factor directly for area or volume.
Claiming congruence from equal angles alone.

Check from memory

What is preserved by similarity?

Corresponding angles and proportional side lengths.

What locus is equidistant from two endpoints?

The perpendicular bisector of the segment joining them.

If area ratio is

49:81

, what is the positive length ratio?

$7:9$ .

Core notes

Congruent figures match in size and shape; similar figures match in shape with a common scale factor. Establish the correspondence before comparing sides, areas or volumes.

Congruence and similarity

Enlargement, reduction and scale drawings

An enlargement or reduction multiplies every length from the centre by one scale factor. A scale drawing represents actual dimensions proportionally, so convert units before applying the scale.

Bisectors and constructions

Area and volume scale factors

If corresponding lengths scale by $k$ , areas scale by $k^2$ and volumes by $k^3$ . Take the appropriate root when working backwards from an area or volume ratio.

Relationship

Use

\text{area ratio}=k^2

Convert a linear scale factor into an area ratio.

\text{volume ratio}=k^3

Convert a linear scale factor into a volume ratio.

Worked examples

Example 1: Two similar solids have corresponding lengths in ratio $2:5$ . Find their volume ratio.

The linear scale factor is $\frac{2}{5}$ .
Cube both terms for volume.
Calculate $2^3:5^3=8:125$ .

Answer: The volume ratio is $8:125$ .

Example 2: A model uses scale $1:200$ . A wall measures $7.5\text{ cm}$ on the model. Find its actual length in metres.

Multiply by $200$ : $7.5\times200=1500\text{ cm}$ .
Convert centimetres to metres.
Calculate $1500\div100=15\text{ m}$ .

Answer: The actual wall is $15\text{ m}$ long.

Exam traps and retrieval check

Avoid these traps

Pairing non-corresponding sides.
Using the linear scale factor directly for area or volume.
Claiming congruence from equal angles alone.

Check from memory

What is preserved by similarity?

Corresponding angles and proportional side lengths.

What locus is equidistant from two endpoints?

The perpendicular bisector of the segment joining them.

If area ratio is

49:81

, what is the positive length ratio?

$7:9$ .

G2: Congruence and similarity

Core notes

Congruence and similarity

Enlargement, reduction and scale drawings

Bisectors and constructions

Area and volume scale factors

Formulae and methods

Worked examples

Example 1: Two similar solids have corresponding lengths in ratio $2:5$ . Find their volume ratio.

Example 2: A model uses scale $1:200$ . A wall measures $7.5\text{ cm}$ on the model. Find its actual length in metres.

Chapter checkpoint

Exam traps and retrieval check

Avoid these traps

Check from memory

Official K310 coverage

G2: Congruence and similarity

Core notes

Congruence and similarity

Enlargement, reduction and scale drawings

Bisectors and constructions

Area and volume scale factors

Formulae and methods

Worked examples

Example 1: Two similar solids have corresponding lengths in ratio $2:5$ . Find their volume ratio.

Example 2: A model uses scale $1:200$ . A wall measures $7.5\text{ cm}$ on the model. Find its actual length in metres.

Chapter checkpoint

Exam traps and retrieval check

Avoid these traps

Check from memory

Official K310 coverage

Relationship	Use
$\text{area ratio}=k^2$	Convert a linear scale factor into an area ratio.
$\text{volume ratio}=k^3$	Convert a linear scale factor into a volume ratio.

Core notes

Congruence and similarity

Enlargement, reduction and scale drawings

Bisectors and constructions

Area and volume scale factors

Formulae and methods

Worked examples

Example 1: Two similar solids have corresponding lengths in ratio 2:52:52:5. Find their volume ratio.

Example 2: A model uses scale 1:2001:2001:200. A wall measures 7.5 cm7.5\text{ cm}7.5 cm on the model. Find its actual length in metres.

Chapter checkpoint

Exam traps and retrieval check

Avoid these traps

Check from memory

Official K310 coverage

Core notes

Congruence and similarity

Enlargement, reduction and scale drawings

Bisectors and constructions

Area and volume scale factors

Formulae and methods

Worked examples

Example 1: Two similar solids have corresponding lengths in ratio 2:52:52:5. Find their volume ratio.

Example 2: A model uses scale 1:2001:2001:200. A wall measures 7.5 cm7.5\text{ cm}7.5 cm on the model. Find its actual length in metres.

Chapter checkpoint

Exam traps and retrieval check

Avoid these traps

Check from memory

Official K310 coverage

Example 1: Two similar solids have corresponding lengths in ratio $2:5$ . Find their volume ratio.

Example 2: A model uses scale $1:200$ . A wall measures $7.5\text{ cm}$ on the model. Find its actual length in metres.

Example 1: Two similar solids have corresponding lengths in ratio $2:5$ . Find their volume ratio.

Example 2: A model uses scale $1:200$ . A wall measures $7.5\text{ cm}$ on the model. Find its actual length in metres.