Question

quadrilateral opqr is dilated by a scale factor of \\(\frac{3}{4}\\) to form quadrilateral opqr. what is the measure of side pq? \
(image of quadrilateral opqr with sides rq = 47, qp = 21, po = 21, or = 47)

Explanation:

Step1: Recall dilation property

In a dilation, the ratio of the length of a side of the image to the length of the corresponding side of the original figure is equal to the scale factor. Let the length of \( PQ \) be \( x \) and the length of \( P'Q' \) be \( 21 \). The scale factor \( k=\frac{3}{4} \), and we know that \( \text{Length of image side} = k\times\text{Length of original side} \), so \( P'Q'=\frac{3}{4}\times PQ \).

Step2: Solve for \( PQ \)

We have \( 21 = \frac{3}{4}x \). To solve for \( x \), multiply both sides by \( \frac{4}{3} \). So \( x = 21\times\frac{4}{3} \).

Step3: Calculate the value

\( 21\times\frac{4}{3}=\frac{21\times4}{3}=7\times4 = 28 \)? Wait, no, wait. Wait, maybe I mixed up. Wait, dilation: if the original is OPQR and the image is O'P'Q'R', then the scale factor is \( \frac{3}{4} \), so \( \text{Image side} = \text{scale factor} \times \text{Original side} \). Wait, but in the diagram, \( P'Q' = 21 \), so \( 21=\frac{3}{4}\times PQ \), so \( PQ = 21\div\frac{3}{4}=21\times\frac{4}{3}=28 \)? Wait, but wait, maybe I got the scale factor direction wrong. Wait, dilation by scale factor \( \frac{3}{4} \) means the image is smaller than the original? Wait, no, if the scale factor is less than 1, the image is a reduction. Wait, but in the diagram, \( P'Q' = 21 \), so original \( PQ \) should be larger? Wait, no, wait, maybe I had it reversed. Let's re - express: Let \( \text{Original side} = PQ \), \( \text{Image side} = P'Q' \). Then \( P'Q'=k\times PQ \), where \( k = \frac{3}{4} \). So \( 21=\frac{3}{4}\times PQ \), so \( PQ=\frac{21\times4}{3}=28 \). Wait, but let's check with another side. The other sides: \( R'Q' = 47 \), so \( RQ=\frac{47\times4}{3}\)? No, that can't be, because 47 is not divisible by 3. Wait, maybe I misidentified the corresponding sides. Wait, looking at the diagram, \( O'P' = 21 \), \( P'Q' = 21 \)? Wait, no, the diagram shows \( O'P' = 21 \), \( P'Q' = 21 \)? Wait, no, the diagram: \( O'P' = 21 \), \( P'Q' = 21 \), \( Q'R' = 47 \), \( R'O' = 47 \). Wait, maybe \( PQ \) corresponds to \( P'Q' \), and the scale factor is \( \frac{3}{4} \), so \( P'Q'=\frac{3}{4}PQ \), so \( PQ=\frac{P'Q'}{\frac{3}{4}}=P'Q'\times\frac{4}{3} \). Since \( P'Q' = 21 \), then \( PQ = 21\times\frac{4}{3}=28 \). Wait, but let's check the calculation again: \( 21\div3 = 7 \), \( 7\times4 = 28 \). Yes.

Answer:

\( \boldsymbol{28} \)