Question

the triangle uvw is a dilation of the triangle uvw. 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: Identify corresponding side lengths

Let's consider the distance from \(U\) to \(V\) and \(U'\) to \(V'\). The coordinates of \(U(- 5,0)\), \(V(0,-10)\), \(U'(0,0)\) and \(V'(0, - 2)\). The length of \(UV\) using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) is \(\sqrt{(0 + 5)^2+(-10-0)^2}=\sqrt{25 + 100}=\sqrt{125}=5\sqrt{5}\). The length of \(U'V'\) is \(\vert-2-0\vert = 2\).

Step2: Calculate the scale - factor

The scale factor \(k\) of a dilation is given by the ratio of the length of a side in the image to the length of the corresponding side in the pre - image. Let's use another approach. If we consider the vertical distance from \(V\) to the \(x\) - axis (\(y = 0\)) which is \(10\) units and the vertical distance from \(V'\) to the \(x\) - axis which is \(2\) units. The scale factor \(k=\frac{\text{length of side in image}}{\text{length of corresponding side in pre - image}}\). So \(k=\frac{2}{10}=\frac{1}{5}\).

Answer:

\(\frac{1}{5}\)