if the ratio of corresponding sides is [tex]\( \frac{1}{2} \), the ratio of the areas is \( \left(\frac{1}{2}\right)^2 = \frac{1}{4} \). Given that the area of quadrilateral QRST is \( \frac{525}{2} \) in^2, the area of WXYZ is \( \frac{525}{2} \times \frac{1}{4} = \frac{525}{8} \) in^2[/tex]. So, statement E is true. Therefore, statements A, B, C, D, and E are all true.
To determine the true statements, let's recall some properties of similar figures:
1. Corresponding angles in similar figures are congruent.
2. Corresponding sides in similar figures are proportional.
Given that quadrilateral WXYZ is similar to quadrilateral QRST, we can use these properties to evaluate the statements.
A. WZ= 15/2 in.
Since WXYZ is similar to QRST, the sides of WXYZ are proportional to the sides of QRST. If WX=5 in., and the corresponding side in QRST is 10 in., then [tex]\( WZ = \frac{5}{10} \times 15 = \frac{15}{2} \)[/tex] in. So, statement A is true.
B. M
The midpoint of a side in a similar figure corresponds to the midpoint of the corresponding side in the other figure. Therefore, if M is the midpoint of QR in QRST, then M is also the midpoint of WZ in WXYZ. So, statement B is true.
C. ZY=25in.
Since WXYZ is similar to QRST, the sides of WXYZ are proportional to the sides of QRST. If QR=20 in., then [tex]\( ZY = \frac{5}{10} \times 50 = 25 \)[/tex]in. So, statement C is true.
D. M
As stated previously, M is the midpoint of WZ, so statement D is true.
E. The area of quadrilateral WXYZ is [tex]525/8 in^2.[/tex]
Since the area of similar figures is proportional to the square of the scale factor, if the ratio of corresponding sides is [tex]\( \frac{1}{2} \), the ratio of the areas is \( \left(\frac{1}{2}\right)^2 = \frac{1}{4} \). Given that the area of quadrilateral QRST is \( \frac{525}{2} \) in^2, the area of WXYZ is \( \frac{525}{2} \times \frac{1}{4} = \frac{525}{8} \) in^2[/tex]. So, statement E is true.
Therefore, statements A, B, C, D, and E are all true.
