Exercise 39 Page 214

The variables are expressions for the right side and the top angle in the triangles. To find $x,$ we compare the right sides and to find $y$ we must compare the top angles.

Scale Factor

To find

x

we must calculate the scale factor of the dilation. The length between center

C

and the blue figure is

2

and the length between

C

and the red figure is

6 .

The scale factor

k

can then be calculated by dividing the image length, from the red figure, with the preimage length, from the blue figure.

k = \frac{image length}{preimage length} = \frac{6}{2} = 3

Right side

We should now compare the expressions for the right sides of the two triangles.

Blue figure : Red figure : Right side x + 1 2 x + 8

The connection between the blue and the red figure is that the sides of the blue figure are multiplied by

3

to get the red figure. Therefore, if we multiply the expression of the right side of the blue figure by

3,

we will get the right side of the red figure.

(x + 1) \cdot 3 = 2 x + 8

We can now solve this equation for

x .

(x + 1) \cdot 3 = 2 x + 8

CommutativePropMult

Commutative Property of Multiplication

3 \cdot (x + 1) = 2 x + 8

▼

Solve for

x

Distr

Distribute $3$

3 x + 3 = 2 x + 8

SubEqn

$LHS - 3 = RHS - 3$

3 x = 2 x + 5

SubEqn

$LHS - 2 x = RHS - 2 x$

x = 5

The variable

x

has the value

5 .

Top angle

We should now find

y

by comparing the expressions for the top angles. We should not use the scale factor when comparing angles since they do not change after a dilation. An angle will stay the same even if you change the length of its legs. Therefore, we should put the expressions equal to each other.