Question

question 7 (2 pts) quadrilateral qrst is similar to quadrilateral uvwx. find the measure of side wx. figures are not drawn to scale. diagrams of two quadrilaterals, qrst with sides ts = 18.8 and rs = 42, and uvwx with side vw = 21, and a box for the answer and a submit answer button

Explanation:

Step1: Determine the scale factor

Since the quadrilaterals are similar, the ratio of corresponding sides is equal. First, find the scale factor by dividing the length of \( VR \) (or \( QS \) corresponding side) in the smaller quadrilateral by the length of \( TS \) (or \( QR \) corresponding side) in the larger quadrilateral. The length of \( TS \) (larger) is 42, and the length of \( VR \) (smaller) is 21. So the scale factor \( k=\frac{21}{42}=\frac{1}{2} \).

Step2: Find the length of \( WX \)

The corresponding side to \( WX \) in the larger quadrilateral is \( TS \) (wait, no, actually, the side \( TS \) in the larger quadrilateral (QRST) has length 18.8? Wait, no, looking at the diagram: QRST has side \( TS = 18.8 \) and \( QS = 42 \)? Wait, no, maybe I mixed up. Wait, quadrilateral QRST: sides are QR, RS, ST, TQ? Wait, the diagram shows QRST with side \( TS = 18.8 \) and \( QS = 42 \)? Wait, no, the lower quadrilateral UVWX has side \( VW = 21 \), and the upper one QRST has side \( QS = 42 \). So the corresponding sides: \( VW \) in UVWX corresponds to \( QS \) in QRST? Wait, no, maybe the sides: QRST and UVWX are similar, so \( ST \) (length 18.8) in QRST corresponds to \( WX \) in UVWX? Wait, no, let's re-express. Let's denote:

In similar quadrilaterals QRST ~ UVWX, so the ratio of corresponding sides is equal. So \( \frac{ST}{WX}=\frac{QS}{VW} \). Wait, \( QS = 42 \), \( VW = 21 \), so the ratio is \( \frac{42}{21}=2 \). Therefore, \( ST = 18.8 \) corresponds to \( WX \), so \( \frac{ST}{WX}=2 \), so \( WX=\frac{ST}{2} \).

Wait, \( ST = 18.8 \), so \( WX=\frac{18.8}{2}=9.4 \)? Wait, no, wait: if QRST is similar to UVWX, then the ratio of QRST to UVWX is \( \frac{QS}{VW}=\frac{42}{21}=2 \), so QRST is twice as big as UVWX. Therefore, the side \( ST \) in QRST (length 18.8) corresponds to \( WX \) in UVWX, so \( WX=\frac{ST}{2}=\frac{18.8}{2}=9.4 \). Wait, but let's check again.

Wait, maybe the sides: in QRST, the side with length 42 is corresponding to the side with length 21 in UVWX. So the scale factor from QRST to UVWX is \( \frac{21}{42}=\frac{1}{2} \). Therefore, the side \( ST \) in QRST (length 18.8) will correspond to \( WX \) in UVWX, so \( WX = ST\times\frac{1}{2}=18.8\times\frac{1}{2}=9.4 \).

Answer:

\( 9.4 \)