Question

pentagon abcde is dilated according to the rule $d_{o,3}(x,y)$ to create the image pentagon $abcde$, which is shown on the graph. what are the coordinates of point a of the pre - image? $(-1,1)$ $(-1,2)$ $(-9,6)$ $(-9,18)$

Explanation:

Step1: Recall the dilation formula

The dilation rule \(D_{O,3}(x,y)=(3x,3y)\) (center at the origin \(O\) with scale factor \(k = 3\)). If \((x',y')\) is the image after dilation, then \(x'=3x\) and \(y' = 3y\) (where \((x,y)\) is the pre - image). We need to find \((x,y)\) given \((x',y')\) of the image.

Step2: Find the coordinates of the image point \(A'\)

From the graph (assuming standard grid - based coordinate reading), the coordinates of \(A'\) (image point) are \((- 3,6)\).

Step3: Solve for \(x\) and \(y\) (pre - image coordinates)

Since \(x'=3x\) and \(y' = 3y\), then \(x=\frac{x'}{3}\) and \(y=\frac{y'}{3}\). Substituting \(x'=-3\) and \(y' = 6\) into the equations:
For \(x\): \(x=\frac{-3}{3}=-1\)
For \(y\): \(y=\frac{6}{3}=2\)

Answer:

\((-1,2)\)