Question

write the coordinates of the vertices after a dilation with a scale factor of 1/3, centered at the origin.

Explanation:

1. First, identify the original coordinates of the vertices:

• Assume the coordinates of point \(B\) are \((- 4,-8)\), point \(C\) are \((2,-6)\), point \(D\) are \((-2,-3)\), and point \(E\) are \((-8,-5)\).
• The rule for dilation centered at the origin with a scale - factor \(k\) is \((x,y)\to(kx,ky)\). Here \(k = \frac{1}{3}\).

1. Calculate the new coordinates for each vertex:

• For point \(B(-4,-8)\):
• \(x\) - coordinate of the new point \(B'\): \(x'=\frac{1}{3}\times(-4)=-\frac{4}{3}\)
• \(y\) - coordinate of the new point \(B'\): \(y'=\frac{1}{3}\times(-8)=-\frac{8}{3}\)
• So, the coordinates of \(B'\) are \((-\frac{4}{3},-\frac{8}{3})\).
• For point \(C(2,-6)\):
• \(x\) - coordinate of the new point \(C'\): \(x'=\frac{1}{3}\times2=\frac{2}{3}\)
• \(y\) - coordinate of the new point \(C'\): \(y'=\frac{1}{3}\times(-6)= - 2\)
• So, the coordinates of \(C'\) are \((\frac{2}{3},-2)\).
• For point \(D(-2,-3)\):
• \(x\) - coordinate of the new point \(D'\): \(x'=\frac{1}{3}\times(-2)=-\frac{2}{3}\)
• \(y\) - coordinate of the new point \(D'\): \(y'=\frac{1}{3}\times(-3)= - 1\)
• So, the coordinates of \(D'\) are \((-\frac{2}{3},-1)\).
• For point \(E(-8,-5)\):
• \(x\) - coordinate of the new point \(E'\): \(x'=\frac{1}{3}\times(-8)=-\frac{8}{3}\)
• \(y\) - coordinate of the new point \(E'\): \(y'=\frac{1}{3}\times(-5)=-\frac{5}{3}\)
• So, the coordinates of \(E'\) are \((-\frac{8}{3},-\frac{5}{3})\).

Answer:

The coordinates of \(B'\) are \((-\frac{4}{3},-\frac{8}{3})\), the coordinates of \(C'\) are \((\frac{2}{3},-2)\), the coordinates of \(D'\) are \((-\frac{2}{3},-1)\), and the coordinates of \(E'\) are \((-\frac{8}{3},-\frac{5}{3})\).