Question

1. create an icon by rotating the shape $270^\circ$ clockwise about $p$.

Explanation:

Step1: Define rotation rule

A 270° clockwise rotation about a point \(P(x_p,y_p)\) transforms a point \((x,y)\) to \((x_p - (y - y_p), y_p + (x - x_p))\), which is equivalent to a 90° counterclockwise rotation.

Step2: Label shape vertices

Let \(P=(4,3)\) (grid coordinates). Label the parallelogram vertices relative to \(P\):

• Vertex 1: \((3,5)\) → relative: \((-1,2)\)
• Vertex 2: \((5,5)\) → relative: \((1,2)\)
• Vertex 3: \((6,1)\) → relative: \((2,-2)\)
• Vertex 4: \((4,1)\) → relative: \((0,-2)\)

Step3: Apply rotation to vertices

Use the rotation rule on relative coordinates:

• \((-1,2)\) → \((4 - 2, 3 + (-1))=(2,2)\)
• \((1,2)\) → \((4 - 2, 3 + 1)=(2,4)\)
• \((2,-2)\) → \((4 - (-2), 3 + 2)=(6,5)\)
• \((0,-2)\) → \((4 - (-2), 3 + 0)=(6,3)\)

Step4: Plot rotated vertices

Connect the new vertices \((2,2)\), \((2,4)\), \((6,5)\), \((6,3)\) to form the rotated shape.

Answer:

The rotated icon is a parallelogram with vertices at grid coordinates \((2,2)\), \((2,4)\), \((6,5)\), and \((6,3)\), centered about point \(P=(4,3)\). When plotted on the grid, this forms the 270° clockwise rotation of the original shape.