Question

the trapezoid mnop is a dilation of the trapezoid mnop. what is the scale factor of the dilation?simplify your answer and write it as a proper fraction, an improper fraction, or a whole number.

Explanation:

Step1: Find the length of corresponding sides

From the graph, assume the length of side \(OP\) (original) is \(5\) units (from \(x = 4\) to \(x = 5\) along \(y = 0\) is \(1\) unit, but actually, if we consider vertical - like side \(MP\): \(M(0,- 10)\) and \(P(5,0)\), the distance \(MP=\sqrt{(5 - 0)^{2}+(0 + 10)^{2}}=\sqrt{25 + 100}=\sqrt{125}=5\sqrt{5}\); \(M'P'\): \(M'(0,-8)\) and \(P'(4,0)\), the distance \(M'P'=\sqrt{(4 - 0)^{2}+(0 + 8)^{2}}=\sqrt{16 + 64}=\sqrt{80}=4\sqrt{5}\). Or more simply, for vertical - like segments (since dilation is uniform). Take the \(y\) - coordinate difference for a point. For example, for point \(M\) with \(y=-10\) and \(M'\) with \(y = - 8\).

Step2: Calculate the scale factor

The scale factor \(k\) of a dilation is given by the formula \(k=\frac{\text{length of side in image}}{\text{length of side in original}}\). Using the \(y\) - coordinate differences (since dilation is centered at the origin, we can use the ratio of the distances from the origin for corresponding points). If we consider the absolute values of \(y\) - coordinates (for non - origin - centered dilation, we can also use side lengths). Let's use the \(y\) - coordinate of \(M\) (\(y_M=-10\)) and \(y_{M'}=-8\). The scale factor \(k=\frac{\vert y_{M'}\vert}{\vert y_{M}\vert}=\frac{8}{10}=\frac{4}{5}\). Another way: if we consider the \(x\) - coordinate of \(P(5,0)\) and \(P'(4,0)\), the scale factor \(k = \frac{4}{5}\) (since dilation formula for a point \((x,y)\) is \((kx,ky)\) when centered at the origin).

Answer:

$\frac{4}{5}$