Question

1. quadrilateral abcd was dilated with the origin as the center of dilation to create quadrilateral abcd. which rule best represents the dilation that was applied to quadrilateral abcd to create quadrilateral abcd? \\( (x, y) \to (0.4x, 0.4y) \\) \\( (x, y) \to (x + 3, y + 1.5) \\) \\( (x, y) \to (2.5x, 2.5y) \\) \\( (x, y) \to (x - 1.5, y + 3) \\)

Explanation:

Step1: Identify a point on ABCD and A'B'C'D'

Let's take point A. From the graph, assume coordinates of A are \((1, 2)\) (estimating from the grid) and coordinates of A' are \((2.5, 5)\) (since \(1\times2.5 = 2.5\) and \(2\times2.5 = 5\)).

Step2: Check the dilation rule

A dilation with center at origin has the rule \((x,y)\to(kx,ky)\), where \(k\) is the scale factor. For point A \((1,2)\) to A' \((2.5,5)\), \(k=\frac{2.5}{1}=2.5\) and \(k=\frac{5}{2}=2.5\). So the scale factor \(k = 2.5\), so the rule is \((x,y)\to(2.5x,2.5y)\). Also, translation rules (like \(x + 3\) or \(x-1.5\)) are for translation, not dilation. The rule \((x,y)\to(0.4x,0.4y)\) would be a reduction, but A'B'C'D' is larger than ABCD, so it's a dilation with \(k = 2.5\).

Answer:

\((x, y) \to (2.5x, 2.5y)\)