Question

1. rectangle pqrs with vertices p(5,15), q(15,15), r(15,10), and s(5,10): k = 4/5

Explanation:

Step1: Recall Dilation Rule

To dilate a point \((x,y)\) with a scale factor \(k\) centered at the origin (assuming center is origin here, since no other center is given), the new coordinates \((x',y')\) are given by \(x' = kx\) and \(y' = ky\).

Step2: Dilate Point P(5,15)

For point \(P(5,15)\), \(x = 5\), \(y = 15\), \(k=\frac{4}{5}\).
\(x'=\frac{4}{5}\times5 = 4\), \(y'=\frac{4}{5}\times15 = 12\). So \(P'(4,12)\).

Step3: Dilate Point Q(15,15)

For point \(Q(15,15)\), \(x = 15\), \(y = 15\).
\(x'=\frac{4}{5}\times15 = 12\), \(y'=\frac{4}{5}\times15 = 12\). So \(Q'(12,12)\).

Step4: Dilate Point R(15,10)

For point \(R(15,10)\), \(x = 15\), \(y = 10\).
\(x'=\frac{4}{5}\times15 = 12\), \(y'=\frac{4}{5}\times10 = 8\). So \(R'(12,8)\).

Step5: Dilate Point S(5,10)

For point \(S(5,10)\), \(x = 5\), \(y = 10\).
\(x'=\frac{4}{5}\times5 = 4\), \(y'=\frac{4}{5}\times10 = 8\). So \(S'(4,8)\).

Answer:

\(P'(4, 12)\), \(Q'(12, 12)\), \(R'(12, 8)\), \(S'(4, 8)\)