Question

in the graph below, pentagon cdefg is the image of cdefg after a dilation.
what are the scale factor and center of the dilation?

Explanation:

Step1: Identify corresponding points

Take point \(D(-1, - 3)\) and \(D'(-1,3)\).

Step2: Calculate the scale factor

The distance from \(D\) to the \(x -\)axis is \(|-3| = 3\), and the distance from \(D'\) to the \(x -\)axis is \(|3|=3\). But wait, let's use the formula for dilation. If \((x,y)\) is dilated to \((x',y')\) with center \((a,b)\) and scale factor \(k\), for a vertical dilation (since \(x\) - coordinate of \(D\) and \(D'\) is the same), \(y' - b=k(y - b)\). Let's assume the center of dilation is on the \(x -\)axis (\(y = 0\)). Then \(y'=ky\). Substituting \(y=-3\) and \(y' = 3\), we get \(3=k\times(- 3)\), so \(k=-1\). But let's check another point. Take \(E(0,-5)\) and \(E'(5,0)\). Using the distance formula, the length of \(DE\): \(d_{DE}=\sqrt{(-1 - 0)^2+(-3+5)^2}=\sqrt{1 + 4}=\sqrt{5}\). The length of \(D'E'\): \(d_{D'E'}=\sqrt{(-1 - 5)^2+(3 - 0)^2}=\sqrt{36 + 9}=\sqrt{45}=3\sqrt{5}\). The scale factor \(k=\frac{d_{D'E'}}{d_{DE}}=\frac{3\sqrt{5}}{\sqrt{5}} = 3\). Wait, no. Wait, actually, if we consider the transformation of coordinates. Let's assume the center of dilation is \((x_0,y_0)\). For a point \((x,y)\) and its image \((x',y')\) after dilation \(x'=x_0+k(x - x_0)\) and \(y'=y_0+k(y - y_0)\). Let's take \(D(-1,-3)\) and \(D'(-1,3)\). \(x\) - coordinate: \(-1=x_0+k(-1 - x_0)\). \(y\) - coordinate: \(3=y_0+k(-3 - y_0)\). If \(x_0=-1\) (since \(x\) - coordinate of \(D\) and \(D'\) is the same), then for \(y\) - coordinate: \(3=y_0+k(-3 - y_0)\). Let's take \(E(0,-5)\) and \(E'(5,0)\). If \(x_0=-1\), then \(x'=-1+k(x + 1)\). For \(x = 0\), \(x'=5\), so \(5=-1+k(0 + 1)\), \(k = 6\) (wrong). Wait, another approach. The center of dilation is the intersection of lines connecting corresponding points. Connect \(D(-1,-3)\) to \(D'(-1,3)\) (vertical line \(x=-1\)). Connect \(E(0,-5)\) to \(E'(5,0)\). The equation of the line through \((0,-5)\) and \((5,0)\) is \(y=x - 5\). The intersection of \(x=-1\) and \(y=x - 5\) is \((-1,-6)\). Let's check the scale factor. Take \(D(-1,-3)\) and \(D'(-1,3)\). The distance from \(D\) to \((-1,-6)\) is \(|-3+6| = 3\), the distance from \(D'\) to \((-1,-6)\) is \(|3 + 6|=9\). Scale factor \(k=\frac{9}{3}=3\).

Answer:

The scale factor is \(3\) and the center of dilation is \((-1,-6)\)