Question

triangle abc has vertices with coordinates a(2, 5), b(4, 2), and c(2, 2). dilate △abc on the coordinate plane using point c as the center of dilation and a scale factor of 2 to form △abc

Explanation:

Step1: Recall dilation formula

For a dilation with center \((x_0,y_0)\) and scale - factor \(k\), the formula for the coordinates of a point \((x,y)\) after dilation is \((x',y')=(x_0 + k(x - x_0),y_0 + k(y - y_0))\). Here, the center of dilation \(C(x_0,y_0)=(2,2)\) and \(k = 2\).

Step2: Find coordinates of \(A'\)

For point \(A(2,5)\), using the dilation formula:
\(x'=2+2\times(2 - 2)=2\)
\(y'=2+2\times(5 - 2)=2 + 6=8\)
So \(A'=(2,8)\).

Step3: Find coordinates of \(B'\)

For point \(B(4,2)\), using the dilation formula:
\(x'=2+2\times(4 - 2)=2+4 = 6\)
\(y'=2+2\times(2 - 2)=2\)
So \(B'=(6,2)\).
\(C\) is the center of dilation, so \(C'=(2,2)\) (a point at the center of dilation does not change).

Answer:

The coordinates of \(\triangle A'B'C'\) are \(A'(2,8)\), \(B'(6,2)\), and \(C'(2,2)\)