Question

polygon abcd is dilated by a scale factor of \\( \frac { 1 } { 2 } \\) centered at the origin.
which two of the following coordinates belong to the image?
select two correct answers.
a. \\( \square ( 0,0 ) \\)
b. \\( \square ( 4,0 ) \\)
c. \\( \square ( 2, - 2 ) \\)
d. \\( \square ( 2,0 ) \\)
e. \\( \square ( 1,2 ) \\)

Explanation:

Step1: Recall the dilation formula

If a point \((x,y)\) is dilated by a scale factor \(k\) centered at the origin, the new point is \((kx,ky)\). Here \(k = \frac{1}{2}\).

Step2: Find the original coordinates

Assume the original coordinates of the polygon vertices (by observing the graph): Let's assume some original points. For example, if there is a point at \((0,0)\) (origin - dilation of origin by any scale factor centered at origin is \((0,0)\)). If there is a point at \((4,0)\), after dilation \(x'=\frac{1}{2}\times4 = 2,y'=\frac{1}{2}\times0=0\) (gives \((2,0)\)). If there is a point at \((2, - 4)\), after dilation \(x'=\frac{1}{2}\times2 = 1,y'=\frac{1}{2}\times(-4)=-2\) (not in options). If there is a point at \((2,4)\), after dilation \(x'=\frac{1}{2}\times2 = 1,y'=\frac{1}{2}\times4 = 2\) (not in options for dilation from a wrong assumed original). But for the origin \((0,0)\): \(x'=\frac{1}{2}\times0 = 0,y'=\frac{1}{2}\times0=0\) (gives \((0,0)\)) and for a point that was originally at \((4,0)\) (after dilation \((2,0)\))

Answer:

A. \((0,0)\), D. \((2,0)\)