Question

write the coordinates of the vertices after a dilation with a scale factor of \\(\frac{1}{3}\\), centered at the origin.
(image of a coordinate grid with a rectangle having vertices a, b, c, d. then boxes for coordinates of a, b, c, d after dilation)

Explanation:

Step1: Find the original coordinates

From the graph, \(A(-9,-9)\), \(B(3,-9)\), \(C(3,6)\), \(D(-9,6)\)

Step2: Apply the dilation formula

For a dilation centered at the origin with scale factor \(k = \frac{1}{3}\), the formula is \((x,y)\to(kx,ky)\)

• For point \(A\):

\(x=-9\), \(y = -9\), \(k=\frac{1}{3}\)
\(x'=\frac{1}{3}\times(-9)=-3\), \(y'=\frac{1}{3}\times(-9)=-3\)

• For point \(B\):

\(x = 3\), \(y=-9\), \(k=\frac{1}{3}\)
\(x'=\frac{1}{3}\times3 = 1\), \(y'=\frac{1}{3}\times(-9)=-3\)

• For point \(C\):

\(x = 3\), \(y = 6\), \(k=\frac{1}{3}\)
\(x'=\frac{1}{3}\times3=1\), \(y'=\frac{1}{3}\times6 = 2\)

• For point \(D\):

\(x=-9\), \(y = 6\), \(k=\frac{1}{3}\)
\(x'=\frac{1}{3}\times(-9)=-3\), \(y'=\frac{1}{3}\times6=2\)

Answer:

\(A'(-3,-3)\), \(B'(1,-3)\), \(C'(1,2)\), \(D'(-3,2)\)