Question

the graph shows quadrilateral abcd. quadrilateral abcd is the image of abcd after a dilation by a scale factor of \\( \frac{1}{2} \\) centered at the origin. what is the perimeter of quadrilateral abcd? units

Explanation:

Step1: Find the lengths of the sides of quadrilateral \(ABCD\)

• For \(AD\): Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), \(A(-7,6)\), \(D(-6,9)\)

\(AD=\sqrt{(-6+7)^2+(9 - 6)^2}=\sqrt{1 + 9}=\sqrt{10}\)

• For \(AB\): \(A(-7,6)\), \(B(-2,2)\)

\(AB=\sqrt{(-2 + 7)^2+(2 - 6)^2}=\sqrt{25+16}=\sqrt{41}\)

• For \(BC\): \(B(-2,2)\), \(C(-2,5)\)

\(BC=\sqrt{(-2+2)^2+(5 - 2)^2}=3\)

• For \(CD\): \(C(-2,5)\), \(D(-6,9)\)

\(CD=\sqrt{(-6 + 2)^2+(9 - 5)^2}=\sqrt{16 + 16}=\sqrt{32}=4\sqrt{2}\)

Step2: Calculate the perimeter of \(ABCD\)

\(P_{ABCD}=AD + AB+BC + CD=\sqrt{10}+\sqrt{41}+3 + 4\sqrt{2}\approx3.16+6.40+3+5.66 = 18.22\)

Step3: Use the property of dilation

If the scale factor of dilation is \(k=\frac{1}{2}\), then the perimeter of the image \(P_{A'B'C'D'}=k\times P_{ABCD}\)
\(P_{A'B'C'D'}=\frac{1}{2}\times18.22 = 9.11\approx9\) (rounded to the nearest whole number)

Answer:

\(9\)