Question

defg has been dilated to defg. use a pair of corresponding sides to determine the scale factor. options: a) 3/2, b) 2/3, c) 1/2, d) 2/1 = 2

Explanation:

Step1: Identify corresponding sides

First, we find the length of a side in the original figure \(DEFG\) and the corresponding side in the dilated figure \(D'E'F'G'\). Let's take the horizontal side \(DG\) and \(D'G'\).

For \(DG\): From the coordinates, \(D\) is at \((-5, 3)\) and \(G\) is at \((-2, 3)\). The length is \(|-2 - (-5)| = 3\) units.

For \(D'G'\): \(D'\) is at \((-4, 2)\) and \(G'\) is at \((-2, 2)\). The length is \(|-2 - (-4)| = 2\) units? Wait, no, maybe better to check vertical or another side. Wait, maybe the vertical or the side \(DE\) and \(D'E'\). Wait, let's check the coordinates again. Wait, maybe the distance between \(D\) and \(E\). Wait, \(D\) is \((-5,3)\), \(E\) is \((-4,-3)\). The length? Wait, no, maybe the horizontal component. Wait, maybe the side \(FG\) and \(F'G'\). Wait, maybe I made a mistake. Let's check the dilated figure. Wait, the scale factor is the ratio of dilated length to original length. Wait, let's take the side \(D'G'\) and \(DG\). Wait, \(D\) is \((-5,3)\), \(G\) is \((-2,3)\): length is \(3\) (since \(x\)-coordinates: \(-2 - (-5) = 3\)). \(D'\) is \((-4,2)\), \(G'\) is \((-2,2)\): length is \(2\)? No, that can't be. Wait, maybe the vertical side. Wait, \(D\) is \((-5,3)\), \(D'\) is \((-4,2)\). Wait, no, maybe the side \(E\) and \(F\). \(E\) is \((-4,-3)\), \(F\) is \((5,-2)\)? Wait, no, the figure is a quadrilateral. Wait, maybe the correct way is to find the ratio of the length of the dilated figure to the original. Wait, the dilated figure is smaller, so scale factor is less than 1. Let's check the coordinates of \(D\) and \(D'\). \(D\) is \((-5,3)\), \(D'\) is \((-4,2)\). Wait, maybe the center of dilation? Wait, the scale factor is \(\frac{\text{length of dilated side}}{\text{length of original side}}\). Let's take the side \(DG\) (original) and \(D'G'\) (dilated). Wait, \(DG\): from \(x=-5\) to \(x=-2\) (same \(y=3\)), so length \(3\). \(D'G'\): from \(x=-4\) to \(x=-2\) (same \(y=2\)), length \(2\)? No, that's not. Wait, maybe the side \(DE\) and \(D'E'\). \(D\) is \((-5,3)\), \(E\) is \((-4,-3)\). The vector from \(D\) to \(E\) is \((1, -6)\). \(D'\) is \((-4,2)\), \(E'\) is \((-3,-2)\). The vector from \(D'\) to \(E'\) is \((1, -4)\). No, that's not. Wait, maybe the horizontal side \(DG\) and \(D'G'\) again. Wait, \(D\) is \((-5,3)\), \(G\) is \((-2,3)\): length \(3\). \(D'\) is \((-4,2)\), \(G'\) is \((-2,2)\): length \(2\). Wait, but \(2/3\) is not an option. Wait, maybe I mixed up original and dilated. Wait, the dilated figure is \(D'E'F'G'\), so original is \(DEFG\), dilated is \(D'E'F'G'\). So scale factor is \(\frac{\text{length of } D'E'F'G'}{\text{length of } DEFG}\). Wait, maybe the side \(D'G'\) and \(DG\): \(D'G'\) length is \(2\), \(DG\) length is \(3\)? No, that would be \(2/3\), but option B is \(2/3\)? Wait, no, option B is \(2/3\)? Wait, the options are A. \(3/2\), B. \(2/3\), C. \(1/2\), D. \(2/1=2\). Wait, maybe I got original and dilated reversed. Wait, if \(D'E'F'G'\) is the dilated figure, then scale factor is dilated length / original length. Wait, maybe the side \(DG\) is original, \(D'G'\) is dilated. Wait, \(DG\) length: \(D(-5,3)\), \(G(-2,3)\): length \(3\). \(D'(-4,2)\), \(G'(-2,2)\): length \(2\). So scale factor is \(2/3\)? But that's option B? Wait, no, wait, maybe the vertical side. Wait, \(D(-5,3)\), \(D'(-4,2)\): no, that's not a side. Wait, maybe the side \(E\) to \(F\). \(E(-4,-3)\), \(F(5,-2)\): length? No, that's complicated. Wait, another approach: the distance from the center of dilation. Wait, the center of dilation is the intersection point of the lines \(DD'\), \(EE'\), \(FF'\), \(GG'\). Let's find the center. The line \(DD'\): from \(D(-5,3)\) to \(D'(-4,2)\). The slope is \((2-3)/(-4+5) = -1/1 = -1\). Equation: \(y - 3 = -1(x + 5)\) → \(y = -x - 2\).

Line \(EE'\): \(E(-4,-3)\) to \(E'(-3,-2)\). Slope: \((-2 + 3)/(-3 + 4) = 1/1 = 1\). Equation: \(y + 3 = 1(x + 4)\) → \(y = x + 1\).

Intersection of \(y = -x - 2\) and \(y = x + 1\): \(-x - 2 = x + 1\) → \(-2x = 3\) → \(x = -1.5\), \(y = -0.5\). Not sure. Alternatively, take the side \(D'G'\) and \(DG\). Wait, \(DG\) is from \((-5,3)\) to \((-2,3)\): length \(3\) (horizontal). \(D'G'\) is from \((-4,2)\) to \((-2,2)\): length \(2\) (horizontal). So scale factor is \(2/3\) (dilated length / original length). So the answer is B. \(2/3\).

Step2: Confirm the ratio

The scale factor is the ratio of the length of a side in the dilated figure (\(D'E'F'G'\)) to the length of the corresponding side in the original figure (\(DEFG\)). For the horizontal side \(DG\) (original) with length \(3\) and \(D'G'\) (dilated) with length \(2\), the scale factor is \(\frac{2}{3}\).

Answer:

B. \(\frac{2}{3}\)