Question

match each point to its image after a rotation of $90^0$ counterclockwise about $(4, -1)$. keyboard help (3, 0) (0, -4) (2, 7) (8, -5) (7, -5) (-4, -3) (8, 3) (3, -2)

Explanation:

To rotate a point \((x, y)\) \(90^\circ\) counterclockwise about a center \((h, k)\), we first translate the point so that the center is at the origin: \((x' = x - h, y' = y - k)\). Then we apply the \(90^\circ\) counterclockwise rotation formula \((x', y') \to (-y', x')\). Finally, we translate back by adding \((h, k)\) to the rotated point: \((x'' = -y' + h, y'' = x' + k)\).

Step 1: Rotate \((3, 0)\) about \((4, -1)\)

• Translate: \(x' = 3 - 4 = -1\), \(y' = 0 - (-1) = 1\)
• Rotate \(90^\circ\) CCW: \((-y', x') = (-1, -1)\)
• Translate back: \(x'' = -1 + 4 = 3\), \(y'' = -1 + (-1) = -2\) → \((3, -2)\)

Step 2: Rotate \((0, -4)\) about \((4, -1)\)

• Translate: \(x' = 0 - 4 = -4\), \(y' = -4 - (-1) = -3\)
• Rotate \(90^\circ\) CCW: \((-y', x') = (3, -4)\)
• Translate back: \(x'' = 3 + 4 = 7\), \(y'' = -4 + (-1) = -5\) → \((7, -5)\)

Step 3: Rotate \((2, 7)\) about \((4, -1)\)

• Translate: \(x' = 2 - 4 = -2\), \(y' = 7 - (-1) = 8\)
• Rotate \(90^\circ\) CCW: \((-y', x') = (-8, -2)\)
• Translate back: \(x'' = -8 + 4 = -4\), \(y'' = -2 + (-1) = -3\) → \((-4, -3)\)

Step 4: Rotate \((8, -5)\) about \((4, -1)\)

• Translate: \(x' = 8 - 4 = 4\), \(y' = -5 - (-1) = -4\)
• Rotate \(90^\circ\) CCW: \((-y', x') = (4, 4)\)
• Translate back: \(x'' = 4 + 4 = 8\), \(y'' = 4 + (-1) = 3\) → \((8, 3)\)

Answer:

• \((3, 0)\) matches \((3, -2)\)
• \((0, -4)\) matches \((7, -5)\)
• \((2, 7)\) matches \((-4, -3)\)
• \((8, -5)\) matches \((8, 3)\)