# Dilation Worksheet

Dilation Worksheet
• Page 1
1.
Which of the following is/are correct?
1. Every dilation has a center.
2. Every dilation has a scale factor.
3. Every dilation increases the size of the picture.
4. Every dilation is a transformation.
 a. 2, 3, and 4 only b. 1, 2 and 4 only c. 1, 2 and 3 only d. All are correct

#### Solution:

Every dilation has a center and a scale factor.

Every dilation is a transformation.

A dilation can either increase or decrease the size of the original picture.

Statements 1, 2 and 4 only are correct.

2.
Select the correct statement/ statements.
1. P is the center of dilation
2. Scale factor of the dilation $x$ maps Q′ to Q
3. QQ′ = PQ ($x$ - 1)

 a. All are correct b. 2 and 3 only c. 1 and 3 only d. 1 and 2 only

#### Solution:

P is the center of dilation.

Scale factor of x maps Q to Q′ and not Q′ to Q.

QQ′ = PQ′ - PQ = x PQ - PQ = PQ (x - 1)

Statements 1 and 3 only are correct.

3.
What is the scale factor that maps square ABCD to square AEFG?
[Given $a$ = 2 and $b$ = 10.]

 a. 6 b. 12 c. 5 d. $\frac{1}{5}$

#### Solution:

Scale factor = AE / AB

= 2+102

= 12 / 2 = 6

4.
Find the scale factor that maps PQRS onto ABCD.

 a. 1.5 b. 3 c. 2 d. 0.5

#### Solution:

Scale factor = DA / SP

= 3 / 6

= 0.5

5.
Find the image of point $P$ (3, 4) under dilation with center (0, 0) with a scale factor of 2.
 a. (- 3, - 4) b. (6, 8) c. (- 6, 8) d. (6, - 8)

#### Solution:

To find the image of a point on the coordinate plane under dilation with center as origin, multiply the coordinates with scale factor.

Image of P (3, 4) is P ′(3 × 2, 4 × 2)

= (6, 8)

6.
Find the image of M(- 6, 12) under dilation with center as origin and with a scale factor of $\frac{5}{6}$.
 a. (- 5, 10) b. (- 7, 14) c. (5 , - 10) d. (7, - 14)

#### Solution:

To find the image of a point on the coordinate plane under a dilation with center as origin, multiply the coordinates with the scale factor.

Image of M(- 6, 12) = (- 6 × 5 / 6, 12 × 5 / 6)

= (- 5, 10)

7.
Find the point of the image of point A after a dilation with center at the origin and scale factor of 2.5.

 a. R b. P c. Q d. M

#### Solution:

Point A is (- 4, 2).

When it is under dilation with a scale factor of 2.5, the image will be (- 4 × 2.5, 2 × 2.5) = (- 10, 5).

This point is shown as P in the picture.

8.
Find the scale factor when Q is dilated to P with center O.

 a. 4 b. $\frac{1}{4}$ c. 3 d. $\frac{1}{3}$

#### Solution:

Q is (12, - 9).

P is (4, - 3).

The coordinates of P are 1 / 3 of those of Q.

Scale factor is 1 / 3.

9.
Select the dilated image of ABCD under a scale factor of 3.

 a. Neither figure 1 nor figure 2 b. Figure 1 c. Figure 2 d. Both Figure 1 and Figure 2

#### Solution:

From the figures, the side AB is 3 times its original length both in figures 1 and 2.

But Figure 1, is not of the same shape of the original. Point C′ is not shown as dilated under a scale facor of 3. So, it is not a transformation.

Figure 2 is of similar size of that of the original. All the points are dilated under a scale factor of 3. It is a transformation.

Figure 2 is the dilation of ABCD under a scale factor of 3.

10.
Points A(1, 1) and B(9, 6) on an XY plane are dilated to A′ and B′ with the center as the origin. The scale factor of A is 4 and that of B is $\frac{1}{3}$. Find the difference in the lengths between the line segments AB and A′ B′.

 a. $\sqrt{63}-\sqrt{13}$ units b. 3$\sqrt{4}$ units c. $\sqrt{89}-\sqrt{5}$ units d. $\frac{4}{3}$units

#### Solution:

A and B are (1, 1) and (9, 6)
[Given.]

A′ = (4, 4)
[Multiply the coordinates by the scale factor.]

B′ = (3, 2)
[Multiply the coordinates by the scale factor.]

AB = [(9 - 1)² + (6 - 1)²]
[Distance formula.]

= 89
[Simplify.]

A′B′ = [(4 - 3)² + (4 - 2)²]
[Distance formula.]

A′B′ = 5
[Simplify.]

AB - A′B′ = 89-5 units
[Steps 5 and 7.]