Question

1. what are the coordinates of \\( \triangle pqr \\) after a dilation with a scale factor of \\( \frac { 2 } { 3 } \\)?
2. determine the transformation that produced the following images:

a)
b)

Explanation:

Step1: Find the coordinates of the original triangle

From the graph, the coordinates of \(\triangle PQR\) are \(P(-6,3)\), \(Q(0,6)\), \(R(6,5)\)

Step2: Apply the dilation formula

The formula for dilation with a scale factor \(k\) centered at the origin is \((x,y)\to(kx,ky)\). Here \(k = \frac{2}{3}\)

• For point \(P(-6,3)\):

\(x=-6,y = 3\), after dilation \(x'=\frac{2}{3}\times(-6)=-4\), \(y'=\frac{2}{3}\times3 = 2\), so \(P'(-4,2)\)

• For point \(Q(0,6)\):

\(x = 0,y=6\), after dilation \(x'=\frac{2}{3}\times0=0\), \(y'=\frac{2}{3}\times6 = 4\), so \(Q'(0,4)\)

• For point \(R(6,5)\):

\(x = 6,y=5\), after dilation \(x'=\frac{2}{3}\times6 = 4\), \(y'=\frac{2}{3}\times5=\frac{10}{3}\), so \(R'(4,\frac{10}{3})\)

Answer:

The coordinates of \(\triangle P'Q'R'\) are \(P'(-4,2)\), \(Q'(0,4)\), \(R'(4,\frac{10}{3})\)