Question

quadrilateral abcd has coordinates a(0, 2), b(3, 4), c(4, 2), and d(3, 0). determine the coordinates of the vertices of abcd after a rotation of 180° about (3, -2)

Explanation:

Step1: Use the rotation formula

The formula for a \(180^{\circ}\) rotation about a point \((h,k)\) is \((x,y)\to(2h - x,2k - y)\). Here \(h = 3\) and \(k=-2\).

Step2: Find \(A'\)

For \(A(0,2)\), \(x = 0\), \(y = 2\).
\(x'=2\times3-0=6\), \(y'=2\times(-2)-2=-4 - 2=-6\). So \(A'(6,-6)\).

Step3: Find \(B'\)

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

Step4: Find \(C'\)

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

Step5: Find \(D'\)

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

Answer:

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