Question

18
quadrilateral abcd has coordinates a (0, 2), b (3, 4), c (4, 2) and d (3, 0). determine the coordinates of the vertices of a b c d after a rotation of 180 degrees about the point (3, -2). * (10 points)
a (2, 0), b (4, 3), c (2, 4) and d (0, 3)
a (-2, 0), b (-4, -3), c (-2, -4) and d (0, -3)
a (6, -6), b (3, -8), c (2, -6) and d (3, -4)
a (0, -2), b (-3, -4), c (-4, -2) and d (-3, 0)

Explanation:

Step1: Use rotation formula

The formula for a \(180^{\circ}\) rotation about a point \((h,k)\) is \((x,y)\to(2h - x,2k - y)\)

Step2: Calculate for point A

For \(A(0,2)\) with \((h = 3,k=-2)\)
\(x'=2\times3-0 = 6\)
\(y'=2\times(-2)-2=-4 - 2=-6\)
So \(A'\) is \((6,-6)\)

Step3: Calculate for point B

For \(B(3,4)\)
\(x'=2\times3 - 3=3\)
\(y'=2\times(-2)-4=-4-4 = -8\)
So \(B'\) is \((3,-8)\)

Step4: Calculate for point C

For \(C(4,2)\)
\(x'=2\times3-4 = 2\)
\(y'=2\times(-2)-2=-4 - 2=-6\)
So \(C'\) is \((2,-6)\)

Step5: Calculate for point D

For \(D(3,0)\)
\(x'=2\times3-3 = 3\)
\(y'=2\times(-2)-0=-4\)
So \(D'\) is \((3,-4)\)

Answer:

C. \(A(6, - 6),B(3, - 8),C(2, - 6)\) and \(D(3, - 4)\)