Question

what are the coordinates of triangle ( abc ) after the dilation?
a city planner is designing a zoomed - in map of a triangular plaza. she starts with triangle ( abc ), with vertices at ( (-4,2) ), ( (6,-2) ), and ( (2,4) ), and creates a dilation centered at the origin with a scale factor of ( 1.5 ) to produce triangle ( abc ).

Explanation:

Step1: Recall the dilation formula

For a dilation centered at the origin \((0,0)\) with scale factor \(k\), if a point \(P(x,y)\) is dilated to \(P'(x',y')\), then \(x' = kx\) and \(y'=ky\). Here \(k = 1.5\).

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

Given \(A(-4,2)\). Using the dilation formula:
\(x'=1.5\times(-4)=-6\) and \(y'=1.5\times2 = 3\). So \(A'(-6,3)\).

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

Given \(B(6,-2)\). Using the dilation formula:
\(x'=1.5\times6 = 9\) and \(y'=1.5\times(-2)=-3\). So \(B'(9,-3)\).

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

Given \(C(2,4)\). Using the dilation formula:
\(x'=1.5\times2=3\) and \(y'=1.5\times4 = 6\). So \(C'(3,6)\).

Answer:

\(A'(-6,3)\), \(B'(9,-3)\), \(C'(3,6)\)