Question

1. triangle abc has vertices a(1, 7), b(3, 2), and c(-2, -2). graph △abc and its image after a rotation of 270° counterclockwise about (-4, 2).

Explanation:

Step1: Translate points to center

For a point \((x,y)\) and center \((h,k)\), translate to \((x - h, y - k)\).

• \(A(1,7)\) → \(A_1(1 - (-4), 7 - 2) = (5,5)\)
• \(B(3,2)\) → \(B_1(3 - (-4), 2 - 2) = (7,0)\)
• \(C(-2,-2)\) → \(C_1(-2 - (-4), -2 - 2) = (2,-4)\)

Step2: Rotate 270° counterclockwise

The rule for 270° counterclockwise rotation is \((x,y) \to (y, -x)\).

• \(A_1(5,5)\) → \(A_2(5, -5)\)
• \(B_1(7,0)\) → \(B_2(0, -7)\)
• \(C_1(2,-4)\) → \(C_2(-4, -2)\)

Step3: Translate back to original center

Translate \((x,y)\) back by \((h,k)\): \((x + h, y + k)\).

• \(A_2(5, -5)\) → \(A'(5 + (-4), -5 + 2) = (1, -3)\)
• \(B_2(0, -7)\) → \(B'(0 + (-4), -7 + 2) = (-4, -5)\)
• \(C_2(-4, -2)\) → \(C'(-4 + (-4), -2 + 2) = (-8, 0)\)

Step4: Graph the triangles

Plot \(A(1,7)\), \(B(3,2)\), \(C(-2,-2)\) for \(\triangle ABC\).
Plot \(A'(1, -3)\), \(B'(-4, -5)\), \(C'(-8, 0)\) for the rotated triangle.

Answer:

The original triangle \(\triangle ABC\) has vertices \(A(1,7)\), \(B(3,2)\), \(C(-2,-2)\). The image after a \(270^\circ\) counterclockwise rotation about \((-4,2)\) has vertices \(A'(1, -3)\), \(B'(-4, -5)\), \(C'(-8, 0)\). (Graphing involves plotting these points and connecting them.)