Question

1. graph the dilated image of quadrilateral abcd with a scale factor of 1.5 and the origin as the center of dilation. list the new ordered pairs abcd. a__ b c d 23. what is the scale factor? 24. what is the scale factor? 25. graph the dilated image of triangle xyz using a scale factor of 0.5 and the origin as a center of dilation. list the new ordered pairs xyz. x y z then translate the image (x + 4, y - 6). list the new ordered pairs xyz. x y z__

Explanation:

Step1: Recall dilation formula

If the center of dilation is the origin \((0,0)\) and the scale - factor is \(k\), the coordinates of a point \((x,y)\) after dilation are \((kx,ky)\).

Step2: Solve problem 22

Assume \(A=(0,2)\), \(B=(2,2)\), \(C=(0,0)\), \(D=(2,0)\).
For point \(A\): \(k = 1.5\), \(A'=(1.5\times0,1.5\times2)=(0,3)\)
For point \(B\): \(B'=(1.5\times2,1.5\times2)=(3,3)\)
For point \(C\): \(C'=(1.5\times0,1.5\times0)=(0,0)\)
For point \(D\): \(D'=(1.5\times2,1.5\times0)=(3,0)\)

Step3: Solve problem 23

Count the distance from a point on the pre - image to the origin and from the corresponding point on the image to the origin. Let's take point \(A(-1,-1)\) on the pre - image and \(A'(-3,-3)\) on the image. The distance from \(A\) to the origin \(d_A=\sqrt{(- 1)^2+(-1)^2}=\sqrt{2}\), the distance from \(A'\) to the origin \(d_{A'}=\sqrt{(-3)^2+(-3)^2}=3\sqrt{2}\). The scale factor \(k=\frac{d_{A'}}{d_A}=3\)

Step4: Solve problem 25

Assume \(X = (-4,2)\), \(Y=(4,0)\), \(Z=(-2,-2)\)
After dilation with \(k = 0.5\):
\(X'=(0.5\times(-4),0.5\times2)=(-2,1)\)
\(Y'=(0.5\times4,0.5\times0)=(2,0)\)
\(Z'=(0.5\times(-2),0.5\times(-2))=(-1,-1)\)
After translation \((x + 4,y-6)\):
\(X''=(-2 + 4,1-6)=(2,-5)\)
\(Y''=(2 + 4,0-6)=(6,-6)\)
\(Z''=(-1 + 4,-1-6)=(3,-7)\)

Answer:

1. \(A'(0,3)\), \(B'(3,3)\), \(C'(0,0)\), \(D'(3,0)\)
2. \(3\)
3. \(X'(-2,1)\), \(Y'(2,0)\), \(Z'(-1,-1)\); \(X''(2,-5)\), \(Y''(6,-6)\), \(Z''(3,-7)\)