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1. Dilations

Warm-up

Three quick questions from earlier lessons.

Discover

A photo is enlarged so every length doubles. What happens to the angles in the photo?

They stay the same
They double

Change the scale factor k. What happens to the angle at A′?

It stays 90°
It is multiplied by k too

Key idea

A dilation stretches from a center.

A dilation with center

O

and scale factor

k > 0

sends each point

P

P'

on ray

OP

with

OP' = k \cdot OP

. About the origin:

(x, y) \to (kx, ky)

. Lengths are multiplied by

k

; angles are unchanged.

Worked example

Dilate $A(2, -1)$ about the origin with $k = 3$ .

Step 1Triangle ABC and the center O.
Step 2Each point slides along its ray from O.
Step 3k = 2: every distance from O doubles; the angles do not change.

A' = (6, -3)

Multiply both coordinates.

OA' = 3 \cdot OA

Check the distance.

Your turn, guided

Your turn: a segment of 4 becomes 10. The scale factor.

2.5

10 ÷ 4.

Practice ★

Dilate $A(3, 1)$ about the origin with scale factor 2. Tap its image.
Dilate $A(-3, 1)$ about the origin with scale factor 3. Tap its image.
Dilate $A(-3, -1)$ about the origin with scale factor 2. Tap its image.

Watch out!

Practice ★★

A dilation takes a segment of length 3 to a segment of length 11. Find the scale factor.
A dilation takes a segment of length 3 to a segment of length 26. Find the scale factor.

A triangle is dilated with scale factor 5. What happens to its angles?
1. They stay the same
2. They are multiplied by 5
3. They grow by 5 degrees

Exit ticket

Three questions to check you have got it.

Dilate $A(3, 1)$ about the origin with scale factor 2. Tap its image.
Dilate $A(-3, 1)$ about the origin with scale factor 3. Tap its image.
Dilate $A(-3, -1)$ about the origin with scale factor 2. Tap its image.

Summary card

Dilations

$(x, y) \to (kx, ky)$ about the origin; lengths × $k$ , angles kept.

Watch out: multiply, do not add.

2. 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.

3. Triangle similarity

Warm-up

Three quick questions from earlier lessons.

Discover

Two triangles each have angles $40^\circ$ and $60^\circ$ . What is the third angle in each?

$80^\circ$ in both
It depends on the sizes

Key idea

Two angles are enough.

AA: two pairs of equal angles make triangles similar. (Also SSS similarity: all sides proportional; SAS similarity: two sides proportional and the included angles equal.) Then corresponding sides are proportional:

\dfrac{DE}{AB} = \dfrac{EF}{BC} = \dfrac{DF}{AC}

Worked example

$\triangle ABC \sim \triangle DEF$ , $AB = 4$ , $BC = 6$ , $DE = 10$ .

Scale factor

2.5

DE/AB.

EF = 15

EF = 2.5 × BC.

Your turn, guided

Your turn: $AB = 3$ , $AC = 5$ , $DE = 9$ . Find $DF$ .

DF = 15

5 × 3.

Practice ★

Two triangles each have a $76^\circ$ angle and a $50^\circ$ angle. What can you conclude?
1. They are similar (AA)
2. They are congruent
3. Nothing without side lengths

$\triangle ABC \sim \triangle DEF$ with $AB = 8$ , $BC = 7$ and $DE = 25$ . Find $EF$ .Not to scale
$\triangle ABC \sim \triangle DEF$ with $AB = 9$ , $BC = 3$ and $DE = 24$ . Find $EF$ .Not to scale

Watch out!

Practice ★★

Which figures are always similar to each other?
1. Any two squares (for example sides 7 and 14)
2. Any two rectangles
3. Any two isosceles triangles

Exit ticket

Three questions to check you have got it.

$\triangle ABC \sim \triangle DEF$ with $AB = 8$ , $BC = 7$ and $DE = 25$ . Find $EF$ .Not to scale
$\triangle ABC \sim \triangle DEF$ with $AB = 9$ , $BC = 3$ and $DE = 24$ . Find $EF$ .Not to scale
$\triangle ABC \sim \triangle DEF$ with $AB = 7$ , $BC = 9$ and $DE = 16$ . Find $EF$ .Not to scale

Summary card

Triangle similarity

AA (or SSS, SAS similarity) → corresponding sides proportional.

Watch out: AA works for triangles only.

4. The slope of a line

Warm-up

Three quick questions from earlier lessons.

Discover

Two slope triangles sit on the same line: one is 2 across and 3 up, the other 4 across. How far up is the second one?

Change the run. Divide the rise by the run each time.

It is always 1.5
It grows with the triangle

Key idea

Every slope triangle on a line gives the same ratio.

The slope of a line is

\dfrac{\text{vertical change}}{\text{horizontal change}} = \dfrac{y_2 - y_1}{x_2 - x_1}

. All slope triangles on a line are similar, so the slope does not depend on which two points you use. A line that goes down from left to right has a negative slope.

Worked example

Through $(1, 2)$ and $(5, 8)$ .

8 - 2 = 6

Vertical change.

5 - 1 = 4

Horizontal change.

Slope

\tfrac{3}{2}

6 ÷ 4.

Your turn, guided

Your turn: the slope through $(0, 7)$ and $(2, 1)$ .

-3

(1 − 7) ÷ (2 − 0).

Practice ★

Find the slope of the line through $(6, 2)$ and $(9, 8)$ .
Find the slope of the line through $(4, 5)$ and $(9, 7)$ .

A slope triangle on a line has a horizontal side of 6 and a vertical side of 2. Another slope triangle on the same line has a horizontal side of 24. How long is its vertical side?

Watch out!

Practice ★★

Find the slope of the line through $(6, 5)$ and $(8, 0)$ .
Find the slope of the line through $(0, 11)$ and $(7, 6)$ .

Boss problem. At the same moment, a 6 ft post casts a 4 ft shadow and a tree casts a 8 ft shadow. The post with its shadow and the tree with its shadow form similar right triangles.

Exit ticket

Three questions to check you have got it.

Find the slope of the line through $(6, 2)$ and $(9, 8)$ .
Find the slope of the line through $(4, 5)$ and $(9, 7)$ .
Find the slope of the line through $(1, 3)$ and $(6, 5)$ .

Summary card

Slope

Rise over run, the same for every slope triangle on the line.

Watch out: down from left to right means negative.

Answers: Dilations, similarity and slope

1. Dilations

$(6, 2)$
$(-9, 3)$
$(-6, -2)$

$\frac{11}{3}$
$\frac{26}{3}$

They stay the same

$(6, 2)$
$(-9, 3)$
$(-6, -2)$

2. Similar figures and scale factors

$27$
$16$

$100$

$36$
$352$

$27$
$16$
$40$

3. Triangle similarity

They are similar (AA)

$\frac{175}{8}$
$8$

Any two squares (for example sides 7 and 14)

$\frac{175}{8}$
$8$
$\frac{144}{7}$

4. The slope of a line

$2$
$\frac{2}{5}$

$- \frac{5}{2}$
$- \frac{5}{7}$

$\frac{3}{2}$

$2$
$\frac{2}{5}$
$\frac{2}{5}$