Question

4 quadrilateral abcd and quadrilateral abcd are shown on the coordinate grid below. which sequence of transformations could be used to map quadrilateral abcd onto quadrilateral abcd? a a reflection across the y - axis, followed by a dilation centered at the origin with a scale factor of 0.5 b a dilation centered at the origin with a scale factor of 0.5, followed by a reflection across the y - axis c a translation 4 units to the right followed by a dilation centered at the origin with a scale factor of 0.5 d a rotation 180° about the point (0, - 0.5), followed by a dilation centered at the origin with a scale factor of 0.5

Explanation:

Step1: Analyze the size change

First, observe the size of the two quadrilaterals. Quadrilateral \(A'B'C'D'\) is smaller than \(ABCD\). A dilation (a transformation that changes the size of a figure) is involved. If we assume a dilation centered at the origin with a scale factor \(k\). Let's take a side - length example. Suppose the length of a side of \(ABCD\) (e.g., \(AB\) which has a length of \(4\) units) and the corresponding side of \(A'B'C'D'\) (e.g., \(A'B'\) which has a length of \(2\) units). Using the formula for dilation \( (x,y)\to(kx,ky)\), if \(AB\) has endpoints (say \(A(- 3,4)\) and \(B(-1,4)\)) and \(A'B'\) has endpoints (say \(A'(1,2)\) and \(B'(2,2)\)), the scale factor \(k=\frac{1}{2}=0.5\).

Step2: Analyze the reflection

Now, analyze the orientation. If we first consider the \(x\) - coordinates. For a point \(P(x,y)\) in \(ABCD\) and its corresponding point \(P'(x',y')\) in \(A'B'C'D'\). If we first do a dilation centered at the origin with a scale factor of \(0.5\): \( (x,y)\to(0.5x,0.5y)\). For example, if \(A(-3,4)\), after dilation \(A_1(-1.5,2)\). But the actual image of \(A\) is \(A'(1,2)\). The transformation from \((-1.5,2)\) to \((1,2)\) (and similar for other points) is a reflection across the \(y\) - axis. The rule for reflection across the \(y\) - axis is \((x,y)\to(-x,y)\). If we first dilate a point \((x,y)\) by a scale factor \(0.5\) centered at the origin \((x,y)\to(0.5x,0.5y)\) and then reflect across the \(y\) - axis \((0.5x,0.5y)\to(-0.5x,0.5y)\). Let's check another point. Suppose \(D(-3, - 5)\). After dilation \(D_1(-1.5,-2.5)\), after reflection across the \(y\) - axis \(D'(1.5,-2.5)\) (matching the coordinate pattern).

Answer:

C. a dilation centered at the origin with a scale factor of \(0.5\), followed by a reflection across the \(y\) - axis