Question

1. quadrilateral abcd with vertices a(0,6), b(-3,-6), c(-9,-6), and d(-12,-3):

a) dilation with scale factor of 1/3 centered at the origin
b) translation along the vector (-5,-1)
a (0,2)
b (-1,-2)
c (-3,-2)
d (-4,-1)

Explanation:

Step1: Dilate the points

For a dilation with scale factor \(k = \frac{1}{3}\) centered at the origin \((0,0)\), the formula for dilation is \((x,y)\to(kx,ky)\).

• For point \(A(0,6)\): \(A'(0\times\frac{1}{3},6\times\frac{1}{3})=(0,2)\)
• For point \(B(- 3,-6)\): \(B'(-3\times\frac{1}{3},-6\times\frac{1}{3})=(-1,-2)\)
• For point \(C(-9,-6)\): \(C'(-9\times\frac{1}{3},-6\times\frac{1}{3})=(-3,-2)\)
• For point \(D(-12,-3)\): \(D'(-12\times\frac{1}{3},-3\times\frac{1}{3})=(-4,-1)\)

Step2: Translate the dilated points

The translation vector is \(\langle - 5,-1
angle\). The formula for translation is \((x,y)\to(x + a,y + b)\), where \(a=-5\) and \(b = - 1\).

• For \(A'(0,2)\): \(A''(0-5,2 - 1)=(-5,1)\)
• For \(B'(-1,-2)\): \(B''(-1-5,-2 - 1)=(-6,-3)\)
• For \(C'(-3,-2)\): \(C''(-3-5,-2 - 1)=(-8,-3)\)
• For \(D'(-4,-1)\): \(D''(-4-5,-1 - 1)=(-9,-2)\)

Answer:

A. \((0,2)\), B. \((-1,-2)\), C. \((-3,-2)\), D. \((-4,-1)\)