Question

1. quadrilateral abcd has vertices a(-2, 4), b(1, 3), c(2, -3), and d(-3, -1). graph quadrilateral abcd and its image after a rotation of 90° counterclockwise about (-1, 2).

Explanation:

Step1: Recall Rotation Rule

For a 90° counterclockwise rotation about a point \((h,k)\), we first translate the point \((x,y)\) by subtracting \((h,k)\) to get \((x - h, y - k)\), then apply the rotation rule \((x', y') = (- (y - k), x - h)\), and finally translate back by adding \((h,k)\) to get \((x' + h, y' + k)\).

Step2: Rotate Point A(-2,4)

• Translate: \((-2 - (-1), 4 - 2) = (-1, 2)\)
• Rotate: \((-2, -1)\) (since \((x,y)\to(-y,x)\) for 90° CCW)
• Translate back: \((-2 + (-1), -1 + 2) = (-3, 1)\) → \(A'(-3,1)\)

Step3: Rotate Point B(1,3)

• Translate: \((1 - (-1), 3 - 2) = (2, 1)\)
• Rotate: \((-1, 2)\)
• Translate back: \((-1 + (-1), 2 + 2) = (-2, 4)\) → \(B'(-2,4)\)

Step4: Rotate Point C(2,-3)

• Translate: \((2 - (-1), -3 - 2) = (3, -5)\)
• Rotate: \((5, 3)\)
• Translate back: \((5 + (-1), 3 + 2) = (4, 5)\) → \(C'(4,5)\)

Step5: Rotate Point D(-3,-1)

• Translate: \((-3 - (-1), -1 - 2) = (-2, -3)\)
• Rotate: \((3, -2)\)
• Translate back: \((3 + (-1), -2 + 2) = (2, 0)\) → \(D'(2,0)\)

Step6: Graphing

• Plot original points \(A(-2,4)\), \(B(1,3)\), \(C(2,-3)\), \(D(-3,-1)\) and connect them.
• Plot rotated points \(A'(-3,1)\), \(B'(-2,4)\), \(C'(4,5)\), \(D'(2,0)\) and connect them.

Answer:

To graph, plot original quadrilateral with \(A(-2,4)\), \(B(1,3)\), \(C(2,-3)\), \(D(-3,-1)\) and its image with \(A'(-3,1)\), \(B'(-2,4)\), \(C'(4,5)\), \(D'(2,0)\) (graphing involves plotting these points on the coordinate plane and connecting them).