Quadrilateral QRST is dilated by a scale factor of \(\frac{2}{3}\) with the origin as the center of dilation to create pentagon Q’R’S’T’. If (x, y) represents the location of any point on pentagon QRST, which ordered pair represents the location of the corresponding point on quadrilateral Q’R’S’T’?
A. \(\left(\frac{3}{2} x, \frac{3}{2} y\right)\)
B. \(\left(\frac{2}{3} x, \frac{2}{3} y\right)\)
C. \((x + \frac{3}{2}, y + \frac{3}{2})\)
D. \((x + \frac{2}{3}, y + \frac{1}{2})\)
The correct answer is B. (left(frac{2}{3} x, frac{2}{3} yright)).
When a figure is dilated with respect to the origin by a scale factor (k), each point ((x, y)) of the original figure is transformed to ((kx, ky)). In this case, since the scale factor is (frac{2}{3}), the points of quadrilateral QRST will be transformed to points on pentagon Q’R’S’T’ as follows:
1. For point ((x, y)) in quadrilateral QRST,
2. The corresponding point in pentagon Q’R’S’T’ is calculated by multiplying both coordinates by (frac{2}{3}).
So, the corresponding point becomes (left(frac{2}{3} x, frac{2}{3} yright)).