Question

using similar polygons the polygons in each pair are similar. find the missing side length. (2) diagrams of trapezoids with sides labeled 12, 24, 15, 25

Explanation:

Step1: Set up proportion for similar polygons

Since the polygons are similar, the ratios of corresponding sides are equal. Let the missing side be \( y \). The corresponding sides are \( 12 \) and \( 15 \), and \( 24 \) and \( 35 \)? Wait, no, looking at the trapezoids: the top bases are \( 12 \) (first trapezoid) and \( 15 \) (second trapezoid), the non - parallel sides? Wait, the first trapezoid has a non - parallel side \( 24 \) and the second has \( 35 \)? Wait, no, maybe the first trapezoid has sides \( 12 \) (top base), \( 24 \) (leg), and the second has \( 15 \) (top base), \( y \) (leg)? Wait, no, let's re - examine. The two trapezoids are similar, so the ratio of corresponding sides is equal. Let's assume the ratio of the top bases is \( \frac{12}{15}=\frac{4}{5} \), and the ratio of the legs should be the same. So if one leg is \( 24 \) in the first trapezoid and \( y \) in the second, then \( \frac{24}{y}=\frac{12}{15} \). Wait, no, maybe the other way. Wait, the first trapezoid: top base \( 12 \), leg \( 24 \); second trapezoid: top base \( 15 \), leg \( y \). Since they are similar, \( \frac{12}{15}=\frac{24}{y} \)? No, that would be if \( 12 \) corresponds to \( 15 \) and \( 24 \) corresponds to \( y \). Wait, cross - multiply: \( 12y = 24\times15 \). Wait, no, maybe the ratio is \( \frac{15}{12}=\frac{y}{24} \). Let's do that.

Step2: Solve the proportion

If \( \frac{15}{12}=\frac{y}{24} \), cross - multiply: \( 12y=15\times24 \).
First, calculate \( 15\times24 = 360 \).
Then, \( y=\frac{360}{12}=30 \). Wait, that makes sense. Because \( \frac{15}{12}=\frac{5}{4} \), and \( \frac{30}{24}=\frac{5}{4} \).

Answer:

\( y = 30 \)