Question

find the measure of yz if the two figures are similar.
options: 16 in., 14 in., 12 in., 10 in.

Explanation:

Step1: Identify corresponding sides

In similar figures, corresponding sides are proportional. Let's assume \( PR = 12 \) in, \( XZ = 18 \) in, \( QR = 8 \) in, and \( YZ \) is the side we need to find. The ratio of \( XZ \) to \( PR \) is the scale factor.

Step2: Calculate the scale factor

The scale factor \( k=\frac{XZ}{PR}=\frac{18}{12}=\frac{3}{2} \).

Step3: Find \( YZ \) using the scale factor

Since \( YZ \) corresponds to \( QR \), we multiply \( QR \) by the scale factor. So \( YZ = QR\times k = 8\times\frac{3}{2}=12 \)? Wait, no, wait. Wait, maybe I mixed up the correspondence. Wait, let's check again. Wait, maybe \( PR \) and \( XZ \) are corresponding, and \( QR \) and \( YZ \)? Wait, no, maybe the other way. Wait, let's re - examine the figures. Let's say triangle (or quadrilateral) \( PQR \) and \( XYZ \) are similar. So \( PR \) corresponds to \( XZ \), and \( QR \) corresponds to \( YZ \)? Wait, no, maybe \( PQ \) and \( XY \), \( QR \) and \( YZ \), \( PR \) and \( XZ \). Wait, the length of \( PR \) is 12, \( XZ \) is 18. So the ratio of similarity is \( \frac{XZ}{PR}=\frac{18}{12}=\frac{3}{2} \). Then \( QR \) is 8, so \( YZ = QR\times\frac{3}{2}=8\times\frac{3}{2}=12 \)? But wait, the options have 12? Wait, no, wait the options are 16,14,12,10. Wait, maybe I got the correspondence wrong. Wait, maybe \( PR \) is 12, \( XZ \) is 18, so the ratio of the smaller to larger is \( \frac{12}{18}=\frac{2}{3} \). Then \( YZ = QR\div\frac{2}{3}=8\times\frac{3}{2}=12 \)? Wait, but let's check again. Wait, maybe the sides are \( PR = 12 \), \( XZ = 18 \), so the scale factor from the smaller to the larger is \( \frac{18}{12}=\frac{3}{2} \). Then \( QR = 8 \), so \( YZ = 8\times\frac{3}{2}=12 \). Wait, but one of the options is 12. Wait, but let's check the calculation again. \( 18\div12 = 1.5 \), \( 8\times1.5 = 12 \). So \( YZ = 12 \) in.

Answer:

12 in. (The option with 12 in, so the answer is the option with 12 in, e.g., if the option is "12 in." then that's the answer.)