Question

rstu and vxyz are quadrilaterals. given rstu - vxyz, describe a sequence of a rigid motion followed by a dilation with center (0,0) that maps rstu to vxyz.
choose the correct answer below.
a. reflection across y - axis, dilation with center (0,0) and scale factor \\( \frac { 1 } { 2 } \\)
b. reflection across y - axis, dilation with center (0,0) and scale factor 2
c. reflection across x - axis, dilation with center (0,0) and scale factor 2
d. reflection across x - axis, dilation with center (0,0) and scale factor \\( \frac { 1 } { 2 } \\)

Explanation:

Step1: Analyze the orientation and size

• First, observe the orientation of the two quadrilaterals. The pink quadrilateral (RSTU) and the blue quadrilateral (VXYZ) are mirror - images with respect to the y - axis. A reflection across the y - axis changes the sign of the x - coordinate of each point \((x,y)\to(-x,y)\).
• Then, compare the side lengths. Let's assume a side length of the pink quadrilateral (RSTU) is \(l_1\) and the corresponding side length of the blue quadrilateral (VXYZ) is \(l_2\). If we count the number of grid units for corresponding sides, we can see that \(l_2=\frac{1}{2}l_1\). For a dilation with center \((0,0)\) and scale factor \(k\), the formula for the transformation of a point \((x,y)\) is \((x,y)\to(kx,ky)\). Here \(k = \frac{1}{2}\).

Answer:

A. Reflection across y - axis, dilation with center \((0,0)\) and scale factor \(\frac{1}{2}\)