Question

the dashed triangle is the image of the solid triangle. the center of dilation is (6,6). what is the scale factor used to create the dilation? enter your answer as an integer or decimal in the box.

Explanation:

Step1: Find the distance from the center of dilation to a vertex of the solid triangle

Let's take a vertex of the solid triangle. For example, consider the horizontal distance from the center \((6,6)\) to a vertex of the solid triangle on the same horizontal line. The center is at \(x = 6\), and a vertex of the solid triangle on the left - hand side is at \(x=- 10\). The distance \(d_1\) (horizontal distance, since \(y\) - coordinate is the same for the center and the vertex on the same horizontal line of the solid triangle) is \(|6-(-10)|=16\) (using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), here \(y_1 = y_2 = 6\)).

Step2: Find the distance from the center of dilation to the corresponding vertex of the dashed triangle

The corresponding vertex of the dashed triangle on the same horizontal line has \(x = 6\) (center) and the other vertex at \(x = 14\). The distance \(d_2\) (horizontal distance, since \(y\) - coordinate is the same for the center and the vertex on the same horizontal line of the dashed triangle) is \(|14 - 6|=8\) (using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), here \(y_1 = y_2 = 6\)).

Step3: Calculate the scale factor

The scale factor \(k\) of a dilation is given by the formula \(k=\frac{\text{distance from center to a point on the image}}{\text{distance from center to the corresponding point on the pre - image}}\). So \(k=\frac{8}{4}=2\) (we could also use vertical distances. For example, take a vertical line. The distance from the center \((6,6)\) to a vertex of the solid triangle with \(y=-9\) is \(|6-(-9)| = 15\), and the distance from the center \((6,6)\) to the corresponding vertex of the dashed triangle with \(y = 9\) is \(|9 - 6|=3\). But if we use the ratio of the lengths of the corresponding sides (using horizontal or vertical segments for simplicity since the center of dilation is given and we can use one - dimensional distance formula for horizontal/vertical segments), we get the scale factor. Another way: Let \(A\) be a vertex of the solid triangle \((-10,6)\) and \(A'\) be the corresponding vertex of the dashed triangle \((14,6)\). The formula for dilation \((x,y)\to(x_0 + k(x - x_0),y_0 + k(y - y_0))\) where \((x_0,y_0)=(6,6)\). For \(x\) - coordinate: \(x'=6 + k(x - 6)\). Substitute \(x=-10\) and \(x'=14\). Then \(14=6 + k(-10 - 6)\), \(14-6=-16k\), \(8=-16k\) (wrong, we should use the correct formula \((x,y)\to(x_0 + k(x - x_0),y_0 + k(y - y_0))\) for the image \((x',y')\) and pre - image \((x,y)\). The correct formula for scale factor when we know two points (one pre - image and one image) with the same \(y\) (or \(x\)) coordinate: \(k=\frac{x'-x_0}{x - x_0}\) (when \(y'=y\)). Let \(x=-10,x_0 = 6,x'=14\), \(k=\frac{14 - 6}{6-(-10)}=\frac{8}{4}=2\) (using the ratio of the lengths of the segments from the center of dilation).

So the scale factor of the dilation is \(2\).

Answer:

\(2\)