Question

the vertices of △abc are a(5,4), b(10,8), and c(20,0). a dilation centered at the origin with scale factor 4/5 maps △abc onto △abc. what are the coordinates of the vertices of △abc?

Explanation:

Step1: Recall the dilation formula

For a dilation centered at the origin with scale factor \(k\), if a point \(P(x,y)\) is dilated to \(P'(x',y')\), then \(x' = kx\) and \(y'=ky\). Here \(k = \frac{4}{5}\).

Step2: Find the coordinates of \(A'\)

Given \(A(5,4)\), using the dilation formula:
\(x_{A'}=\frac{4}{5}\times5 = 4\)
\(y_{A'}=\frac{4}{5}\times4=\frac{16}{5}=3.2\)
So \(A'(4,3.2)\)

Step3: Find the coordinates of \(B'\)

Given \(B(10,8)\), using the dilation formula:
\(x_{B'}=\frac{4}{5}\times10 = 8\)
\(y_{B'}=\frac{4}{5}\times8=\frac{32}{5} = 6.4\)
So \(B'(8,6.4)\)

Step4: Find the coordinates of \(C'\)

Given \(C(20,0)\), using the dilation formula:
\(x_{C'}=\frac{4}{5}\times20=16\)
\(y_{C'}=\frac{4}{5}\times0 = 0\)
So \(C'(16,0)\)

Answer:

\(A'(4,3.2)\), \(B'(8,6.4)\), \(C'(16,0)\)