Question

3 adnan wants a larger garden. he draws △pqr to show his original garden. he wants to enlarge the garden, so he uses a dilation to form the image △pqr. adnan uses vertex p as the center of dilation. a. write a fraction for each quotient of corresponding side lengths. (\frac{pq}{pq}=\frac{3}{2}), (\frac{qr}{qr}=\frac{3}{2}), (\frac{pr}{pr}=\frac{3}{2}) b. what do your answers to problem 3a tell you about the scale factor adnan used to dilate △pqr to form △pqr? the scale factor is (\frac{3}{2}) c. how do you know that the corresponding angles of △pqr and △pqr are congruent?

Explanation:

Step1: Find side - length ratios

For $\frac{PQ'}{PQ}$, $PQ = 8$ ft and $PQ'=12$ ft, so $\frac{PQ'}{PQ}=\frac{12}{8}=\frac{3}{2}$.
For $\frac{Q'R'}{QR}$, $QR = 10$ ft and $Q'R' = 15$ ft, so $\frac{Q'R'}{QR}=\frac{15}{10}=\frac{3}{2}$.
For $\frac{PR'}{PR}$, $PR = 6$ ft and $PR'=9$ ft, so $\frac{PR'}{PR}=\frac{9}{6}=\frac{3}{2}$.

Step2: Determine the scale factor

Since the ratios of all corresponding side - lengths $\frac{PQ'}{PQ}=\frac{Q'R'}{QR}=\frac{PR'}{PR}=\frac{3}{2}$, the scale factor of the dilation is $\frac{3}{2}$.

Step3: Explain angle congruence

Dilations are a type of similarity transformation. In a similarity transformation, corresponding angles of similar figures are congruent because the shape of the figure is preserved. Only the size changes.

Answer:

a. $\frac{PQ'}{PQ}=\frac{3}{2}$, $\frac{Q'R'}{QR}=\frac{3}{2}$, $\frac{PR'}{PR}=\frac{3}{2}$
b. The scale factor is $\frac{3}{2}$
c. Dilations are similarity transformations, and in similarity transformations, corresponding angles of similar figures are congruent as the shape is preserved while only the size changes.