Question

1. z(3,8) is the image of z after a dilation centered at the origin with a scale factor of 3/2. what are the coordinates of z? 14. h(1.5,-4) is the image of h after a dilation centered at the origin with a scale factor of 1/2. what are the coordinates of h?

Explanation:

Step1: Recall the dilation formula

If a point \(P(x,y)\) is dilated with a scale factor \(k\) centered at the origin, the image \(P'(x',y')\) is given by \(x' = kx\) and \(y'=ky\).

Step2: Solve for problem 13

Given \(Z'(3,8)\) and \(k = \frac{3}{2}\). Let \(Z(x,y)\). Then \(x=\frac{x'}{k}\) and \(y = \frac{y'}{k}\).
Substitute \(x' = 3\) and \(k=\frac{3}{2}\), we get \(x=\frac{3}{\frac{3}{2}}=3\times\frac{2}{3}=2\).
Substitute \(y' = 8\) and \(k=\frac{3}{2}\), we get \(y=\frac{8}{\frac{3}{2}}=8\times\frac{2}{3}=\frac{16}{3}\).

Step3: Solve for problem 14

Given \(H'(1.5,-4)\) and \(k=\frac{1}{2}\). Let \(H(x,y)\). Then \(x = 2x'\) and \(y=2y'\).
Substitute \(x' = 1.5\), we get \(x=2\times1.5 = 3\).
Substitute \(y'=-4\), we get \(y=2\times(-4)=-8\).

Answer:

1. The coordinates of \(Z\) are \((2,\frac{16}{3})\).
2. The coordinates of \(H\) are \((3,-8)\).