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1. Similar figures and scale factors

Warm-up

Three quick questions from earlier lessons.

Discover

A 2 by 3 rectangle is enlarged with scale factor 2. What is the area of the new rectangle?

Key idea

Similar figures: a dilation (and rigid moves) apart.

Two figures are similar if a sequence of rigid transformations and dilations maps one onto the other: corresponding angles are equal and corresponding lengths are in the same ratio

k

. Perimeters scale by

k

, areas by

k^2

(and volumes by

k^3

Worked example

Similar triangles with scale factor 3; the small one has perimeter 12 and area 6.

Perimeter

36

Lengths × 3.

Area

54

Areas × 9.

Your turn, guided

Your turn: scale factor 4, small area 5. The large area.

80

5 × 16.

Practice ★

Two polygons are similar. A side of length 5 corresponds to a side of length 15. Another side of the small polygon has length 9. Find the matching side of the large one.
Two polygons are similar. A side of length 5 corresponds to a side of length 20. Another side of the small polygon has length 4. Find the matching side of the large one.

A polygon has perimeter 20. It is enlarged by a scale factor of 5. Find the new perimeter.

Watch out!

Practice ★★

A shape has area 9. It is enlarged by a scale factor of 2. Find the new area.
A shape has area 22. It is enlarged by a scale factor of 4. Find the new area.

Exit ticket

Three questions to check you have got it.

Two polygons are similar. A side of length 5 corresponds to a side of length 15. Another side of the small polygon has length 9. Find the matching side of the large one.
Two polygons are similar. A side of length 5 corresponds to a side of length 20. Another side of the small polygon has length 4. Find the matching side of the large one.
Two polygons are similar. A side of length 3 corresponds to a side of length 15. Another side of the small polygon has length 8. Find the matching side of the large one.

Summary card

Similar figures

Lengths × $k$ , perimeters × $k$ , areas × $k^2$ .

Watch out: scale by multiplying, and square $k$ for areas.

2. Scale drawings and maps

Warm-up

Three quick questions from earlier lessons.

Discover

On a map, 1 cm stands for 5 km. Two towns are 4 cm apart on the map. How far apart are they?

20 km
0.8 km

Key idea

A scale links drawing lengths to real lengths.

A scale like “1 cm to 5 km” says what one unit on the drawing stands for. Drawing → real: multiply; real → drawing: divide. A scale without units, like 1 to 200, means real lengths are 200 times the drawing lengths in the same unit.

Worked example

Scale 1 to 50; a table is 3 cm long on the drawing.

3 \cdot 50 = 150

Same unit.

1.5

100 cm = 1 m.

Your turn, guided

Your turn: scale 1 inch to 4 feet. A 28-foot wall on the plan (inches).

7

28 ÷ 4.

Practice ★

On a map, 1 cm represents 9 km. Two towns are 6 cm apart on the map. How far apart are they in reality (km)?
On a map, 1 cm represents 16 km. Two towns are 6 cm apart on the map. How far apart are they in reality (km)?

A floor plan uses the scale 1 inch to 4 feet. A wall is 20 feet long. How long is it on the plan (inches)?

Watch out!

Practice ★★

A drawing has a scale of 1 to 450 (no units). A line on it is 11 cm long. How long is the real object in meters?
A drawing has a scale of 1 to 400 (no units). A line on it is 8 cm long. How long is the real object in meters?

Exit ticket

Three questions to check you have got it.

On a map, 1 cm represents 9 km. Two towns are 6 cm apart on the map. How far apart are they in reality (km)?
On a map, 1 cm represents 16 km. Two towns are 6 cm apart on the map. How far apart are they in reality (km)?
On a map, 1 cm represents 5 km. Two towns are 14 cm apart on the map. How far apart are they in reality (km)?

Summary card

Scale drawings

Drawing → real: multiply by the scale; real → drawing: divide.

Watch out: convert units at the end.

3. Scaled copies: lengths and areas

Warm-up

Three quick questions from earlier lessons.

Discover

A 1 cm by 1 cm square is copied with scale factor 3. What is the area of the copy?

9 cm²
3 cm²

Set k = 3. How many times larger is the area?

9 times
3 times

Key idea

Lengths scale by $k$ , areas by $k^2$ .

In a scaled copy with scale factor

k

, every length is multiplied by

k

, angles stay the same, and every area is multiplied by $k^2$ . For a real area from a scale drawing, scale the lengths first, then multiply.

Worked example

A 4 cm² triangle, scale factor 5.

Area

\times 25

k² = 25.

100

cm²

4 · 25.

Your turn, guided

Your turn: area 6 cm², scale factor 2. The copy's area.

24

6 · 4.

Practice ★

A figure has an area of 10 cm². A scaled copy uses a scale factor of 6. What is the area of the copy (cm²)?
A figure has an area of 5 cm². A scaled copy uses a scale factor of 2. What is the area of the copy (cm²)?

A figure of area 10 cm² has a scaled copy of area 90 cm². What is the scale factor?

Watch out!

Practice ★★

On a plan with scale 1 cm to 4 m, a room is 4 cm by 7 cm. What is the room's real area (m²)?
On a plan with scale 1 cm to 4 m, a room is 6 cm by 4 cm. What is the room's real area (m²)?

Boss problem. On a map with scale 1 cm to 60 m, a rectangular park is 2 cm by 12 cm.

Exit ticket

Three questions to check you have got it.

A figure has an area of 10 cm². A scaled copy uses a scale factor of 6. What is the area of the copy (cm²)?
A figure has an area of 5 cm². A scaled copy uses a scale factor of 2. What is the area of the copy (cm²)?
A figure has an area of 17 cm². A scaled copy uses a scale factor of 4. What is the area of the copy (cm²)?

Summary card

Scaling and area

Lengths × $k$ , areas × $k^2$ .

Watch out: scale both sides before multiplying.

Answers: Scale drawings

1. Similar figures and scale factors

$27$
$16$

$100$

$36$
$352$

$27$
$16$
$40$

2. Scale drawings and maps

$54$
$96$

$\frac{99}{2}$
$32$

$54$
$96$
$70$

3. Scaled copies: lengths and areas

$360$
$20$

$448$
$384$

$86400$

$360$
$20$
$272$