Question

dilate figure ( abcd ) by a scale factor of ( \frac{1}{2} ) with the center of dilation at the origin. graph figure ( abcd ).

Explanation:

Step1: Recall the dilation formula

When dilating a point \((x,y)\) with a scale factor \(k\) and center of dilation at the origin \((0,0)\), the formula for the new point \((x',y')\) is \(x'=k\times x\) and \(y' = k\times y\). Here \(k=\frac{1}{2}\).

Step2: Calculate coordinates for \(A'\)

For \(A(4,4)\), \(x'=\frac{1}{2}\times4 = 2\), \(y'=\frac{1}{2}\times4=2\), so \(A'(2,2)\).

Step3: Calculate coordinates for \(B'\)

For \(B(6,2)\), \(x'=\frac{1}{2}\times6 = 3\), \(y'=\frac{1}{2}\times2=1\), so \(B'(3,1)\).

Step4: Calculate coordinates for \(C'\)

For \(C(6,6)\), \(x'=\frac{1}{2}\times6 = 3\), \(y'=\frac{1}{2}\times6=3\), so \(C'(3,3)\).

Step5: Calculate coordinates for \(D'\)

For \(D(4,8)\), \(x'=\frac{1}{2}\times4 = 2\), \(y'=\frac{1}{2}\times8=4\), so \(D'(2,4)\).

Step6: Plot the points

Plot \(A'(2,2)\), \(B'(3,1)\), \(C'(3,3)\), \(D'(2,4)\) on the coordinate - plane and connect them in order \(A'\to B'\to C'\to D'\to A'\) to form the dilated figure \(A'B'C'D'\).

Answer:

The coordinates of the dilated figure \(A'B'C'D'\) are \(A'(2,2)\), \(B'(3,1)\), \(C'(3,3)\), \(D'(2,4)\) which are obtained by applying the dilation formula \( (x',y')=(k x,k y)\) with \(k = \frac{1}{2}\) to the coordinates of \(A(4,4)\), \(B(6,2)\), \(C(6,6)\), \(D(4,8)\) respectively. Then graph the points \(A'\), \(B'\), \(C'\), \(D'\) and connect them.