Question

apply it
1 lillie draws △abc in the coordinate plane. she dilates △abc using a scale factor of $\frac{2}{3}$ with the center of dilation at the origin to form △abc. then she dilates △abc using a scale factor of $\frac{1}{2}$ with a center of dilation at the origin to form △abc. what are the coordinates of △abc? show your work.

Explanation:

Step1: Identify the coordinates of $\triangle ABC$

From the graph, assume $A(-2,-2)$, $B(2,3)$, $C(6,0)$.

Step2: First - dilation

When dilating $\triangle ABC$ with a scale factor of $\frac{2}{3}$ centered at the origin, the rule for dilation is $(x,y)\to(\frac{2}{3}x,\frac{2}{3}y)$.
For point $A(-2,-2)$: $A'(\frac{2}{3}\times(-2),\frac{2}{3}\times(-2)) = A'(-\frac{4}{3},-\frac{4}{3})$
For point $B(2,3)$: $B'(\frac{2}{3}\times2,\frac{2}{3}\times3)=B'(\frac{4}{3},2)$
For point $C(6,0)$: $C'(\frac{2}{3}\times6,\frac{2}{3}\times0)=C'(4,0)$

Step3: Second - dilation

When dilating $\triangle A'B'C'$ with a scale factor of $\frac{1}{2}$ centered at the origin, the rule for dilation is $(x,y)\to(\frac{1}{2}x,\frac{1}{2}y)$
For point $A'(-\frac{4}{3},-\frac{4}{3})$: $A''(\frac{1}{2}\times(-\frac{4}{3}),\frac{1}{2}\times(-\frac{4}{3}))=A''(-\frac{2}{3},-\frac{2}{3})$
For point $B'(\frac{4}{3},2)$: $B''(\frac{1}{2}\times\frac{4}{3},\frac{1}{2}\times2)=B''(\frac{2}{3},1)$
For point $C'(4,0)$: $C''(\frac{1}{2}\times4,\frac{1}{2}\times0)=C''(2,0)$

Answer:

$A''(-\frac{2}{3},-\frac{2}{3})$, $B''(\frac{2}{3},1)$, $C''(2,0)$