Question

the polygons below are similar. find the values of x and y.
a) find the length x.
b) find the length y.

Explanation:

a) Find the length \( x \)

Step 1: Determine the scale factor

Since the polygons are similar, the ratios of corresponding sides are equal. The corresponding sides with known lengths are 18 (from the first polygon) and 10 (from the second polygon). So the scale factor from the first polygon to the second is \( \frac{10}{18}=\frac{5}{9} \), or from the second to the first is \( \frac{18}{10}=\frac{9}{5} \). We can use the side of length 13 in the second polygon and find the corresponding side \( x \) in the first polygon. So we set up the proportion \( \frac{x}{13}=\frac{18}{10} \) (because \( x \) corresponds to 13, and 18 corresponds to 10).

Step 2: Solve for \( x \)

Cross - multiply: \( 10x = 13\times18 \)
\( 10x=234 \)
Divide both sides by 10: \( x=\frac{234}{10} = 23.4 \)

b) Find the length \( y \)

Step 1: Set up the proportion for \( y \)

Using the corresponding sides 18 (first polygon) and 10 (second polygon), and the sides 10.8 (first polygon) and \( y \) (second polygon). The proportion is \( \frac{10.8}{y}=\frac{18}{10} \) (since the ratio of corresponding sides of similar polygons is equal).

Step 2: Solve for \( y \)

Cross - multiply: \( 18y=10.8\times10 \)
\( 18y = 108 \)
Divide both sides by 18: \( y=\frac{108}{18}=6 \)

Answer:

s:
a) \( x = 23.4 \)
b) \( y = 6 \)