Question

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: $\frac{pq}{pq}=$, $\frac{qr}{qr}=$, $\frac{pr}{pr}=$
b. what do your answers to problem 3a tell you about the scale factor adnan used to dilate △pqr to form △pqr?
c. how do you know that the corresponding angles of △pqr and △pqr are congruent?
figure wxyz is a dilation of figure wxyz. label the center of dilation. then write a similarity statement.

Explanation:

Step1: Recall dilation properties

In a dilation, the ratios of corresponding - side lengths are equal to the scale factor. Let the scale factor be \(k\). For \(\triangle PQR\) and \(\triangle PQ'R'\) with center of dilation at \(P\), if the lengths of corresponding sides are \(PQ\) and \(PQ'\), \(QR\) and \(Q'R'\), \(PR\) and \(PR'\), then \(\frac{PQ'}{PQ}=k\), \(\frac{Q'R'}{QR}=k\), \(\frac{PR'}{PR}=k\).

Step2: Answer part a

Since we are not given specific side - lengths, but in general for a dilation of \(\triangle PQR\) to \(\triangle PQ'R'\) with center of dilation \(P\), \(\frac{PQ'}{PQ}\), \(\frac{Q'R'}{QR}\), \(\frac{PR'}{PR}\) are the ratios of corresponding side lengths.

Step3: Answer part b

The equal values of the ratios \(\frac{PQ'}{PQ}\), \(\frac{Q'R'}{QR}\), \(\frac{PR'}{PR}\) from part a tell us that the scale factor \(k\) used to dilate \(\triangle PQR\) to \(\triangle PQ'R'\) is the common value of these ratios. If the ratios are all equal to \(k\), then \(k\) is the factor by which each side of \(\triangle PQR\) is multiplied to get the corresponding side of \(\triangle PQ'R'\).

Step4: Answer part c

Dilations are a type of similarity transformation. In similar triangles (and \(\triangle PQR\sim\triangle PQ'R'\) because of the dilation), corresponding angles are congruent. This is a property of similar figures. When a figure is dilated, the shape of the figure remains the same, only the size changes. So, \(\angle QPR\cong\angle Q'PR'\), \(\angle PQR\cong\angle PQ'R'\), and \(\angle PRQ\cong\angle PR'Q'\) because similar figures have congruent corresponding angles.

For the second part about figure \(WXYZ\) and \(W'X'Y'Z'\):
To find the center of dilation, we look for the point from which the distances to the corresponding vertices of the original and dilated figures are in the same ratio. We draw lines connecting corresponding vertices (\(W\) and \(W'\), \(X\) and \(X'\), \(Y\) and \(Y'\), \(Z\) and \(Z'\)) and find their intersection point, which is the center of dilation.
The similarity statement is \(WXYZ\sim W'X'Y'Z'\).

Answer:

a. \(\frac{PQ'}{PQ}\), \(\frac{Q'R'}{QR}\), \(\frac{PR'}{PR}\)
b. The common value of the ratios in part a is the scale factor of the dilation.
c. Dilations create similar triangles, and similar triangles have congruent corresponding angles.
For figure \(WXYZ\) and \(W'X'Y'Z'\): Center of dilation is the intersection of lines connecting corresponding vertices; Similarity statement: \(WXYZ\sim W'X'Y'Z'\)