Question

the smaller triangle is a dilation of the larger triangle with a center of dilation at (2, - 1)
what is the scale factor of the dilation?
1/3
1/2
2
3

Explanation:

Step1: Select corresponding sides

Choose a side of the larger and smaller triangle. Let's take the vertical sides. The vertical side of the larger triangle from \(y = 1\) to \(y = 5\) has a length of \(5 - 1=4\). The vertical side of the smaller triangle from \(y = 1\) to \(y = 2\) has a length of \(2 - 1 = 1\).

Step2: Calculate scale - factor

The scale factor \(k\) of a dilation is the ratio of the length of a side of the image (smaller triangle) to the length of the corresponding side of the pre - image (larger triangle). So \(k=\frac{\text{length of side of smaller triangle}}{\text{length of side of larger triangle}}=\frac{1}{4/1}=\frac{1}{4}\). But we can also use another way. Consider the distance of a point from the center of dilation \((2,-1)\). Let's take the top - right vertex of the larger triangle \((1,5)\) and the corresponding vertex of the smaller triangle \((1,2)\). The distance of the vertex of the larger triangle from the center of dilation \((2,-1)\) in the \(y\) - direction is \(5-(-1)=6\), and the distance of the vertex of the smaller triangle from the center of dilation \((2,-1)\) in the \(y\) - direction is \(2 - (-1)=3\). The scale factor \(k=\frac{3}{6}=\frac{1}{2}\).

Answer:

\(\frac{1}{2}\)