Question

graph \\( \triangle a b c \\) with vertices \\( a ( - 3, - 6 ) \\), \\( b ( 3,6 ) \\), and \\( c ( 3,3 ) \\) and its image after a dilation centered at the origin with a scale factor of \\( \frac { 1 } { 3 } \\)

Explanation:

Step1: Apply dilation formula

For a dilation centered at the origin with scale factor \(k\), the formula is \((x,y)\to(kx,ky)\). Here \(k = \frac{1}{3}\).
For point \(A(-3,-6)\):
\(x'=\frac{1}{3}\times(-3)= - 1\), \(y'=\frac{1}{3}\times(-6)=-2\), so \(A'(-1,-2)\)

Step2: For point \(B(3,6)\)

\(x'=\frac{1}{3}\times3 = 1\), \(y'=\frac{1}{3}\times6 = 2\), so \(B'(1,2)\)

Step3: For point \(C(3,3)\)

\(x'=\frac{1}{3}\times3=1\), \(y'=\frac{1}{3}\times3 = 1\), so \(C'(1,1)\)

Answer:

The vertices of the image \(\triangle A'B'C'\) are \(A'(-1,-2)\), \(B'(1,2)\) and \(C'(1,1)\)