Question

a triangle has vertices x(1,1),y(5,1),z(3,4). after a dilation, its image has vertices x(3,3),y(15,3),z(9,12). what is the scale factor?
a. 2
b. 1/3
c. 3
d. 1/2

Explanation:

Step1: Recall dilation formula

For a dilation centered at the origin, if the scale - factor is \(k\), the coordinates of a point \((x,y)\) and its image \((x',y')\) are related by \(x' = kx\) and \(y'=ky\). Let's use the \(x\) - coordinates of one of the points.

Step2: Calculate scale - factor using \(x\) - coordinates of point \(X\)

For point \(X(1,1)\) and \(X'(3,3)\), using the formula \(x' = kx\), we substitute \(x = 1\) and \(x'=3\). Then \(3=k\times1\), so \(k = 3\). We can double - check with another point. For point \(Y(5,1)\) and \(Y'(15,3)\), using the \(x\) - coordinates, \(15=k\times5\), solving for \(k\) gives \(k=\frac{15}{5}=3\).

Answer:

C. 3