Question

1. perform the following rotations on quadrilateral abcd.

a. rotate 90° clockwise around the origin. appropriately label the vertices.
b. rotate 180° around (-2,2). appropriately label the vertices.

Explanation:

To solve the rotation problems for quadrilateral \(ABCD\), we first need the coordinates of its vertices. Let's assume the coordinates (from the graph, estimating the grid):

• Let \(A = (-3, 6)\), \(B = (-5, 3)\), \(C = (-4, 2)\), \(D = (-2, 3)\) (adjusted for the grid; actual coordinates may vary slightly, but the method remains the same).

Part (a): Rotate \(90^\circ\) clockwise around the origin

The rule for a \(90^\circ\) clockwise rotation about the origin is:
\((x, y) \to (y, -x)\)

Step 1: Rotate \(A(-3, 6)\)

Apply the rule: \((-3, 6) \to (6, 3)\) (since \(x = -3\), \(y = 6\); new \(x = 6\), new \(y = -(-3) = 3\)).
New vertex: \(A' = (6, 3)\)

Step 2: Rotate \(B(-5, 3)\)

Apply the rule: \((-5, 3) \to (3, 5)\) (new \(x = 3\), new \(y = -(-5) = 5\)).
New vertex: \(B' = (3, 5)\)

Step 3: Rotate \(C(-4, 2)\)

Apply the rule: \((-4, 2) \to (2, 4)\) (new \(x = 2\), new \(y = -(-4) = 4\)).
New vertex: \(C' = (2, 4)\)

Step 4: Rotate \(D(-2, 3)\)

Apply the rule: \((-2, 3) \to (3, 2)\) (new \(x = 3\), new \(y = -(-2) = 2\)).
New vertex: \(D' = (3, 2)\)

Plot \(A'(6, 3)\), \(B'(3, 5)\), \(C'(2, 4)\), \(D'(3, 2)\) and connect them to form the rotated quadrilateral.

Part (b): Rotate \(180^\circ\) around \((-2, 2)\)

The rule for a \(180^\circ\) rotation about a point \((h, k)\) is:
\((x, y) \to (2h - x, 2k - y)\)
Here, \(h = -2\), \(k = 2\), so the rule becomes:
\((x, y) \to (2(-2) - x, 2(2) - y) = (-4 - x, 4 - y)\)

Step 1: Rotate \(A(-3, 6)\)

Apply the rule: \((-3, 6) \to (-4 - (-3), 4 - 6) = (-1, -2)\)
New vertex: \(A'' = (-1, -2)\)

Step 2: Rotate \(B(-5, 3)\)

Apply the rule: \((-5, 3) \to (-4 - (-5), 4 - 3) = (1, 1)\)
New vertex: \(B'' = (1, 1)\)

Step 3: Rotate \(C(-4, 2)\)

Apply the rule: \((-4, 2) \to (-4 - (-4), 4 - 2) = (0, 2)\)
New vertex: \(C'' = (0, 2)\)

Step 4: Rotate \(D(-2, 3)\)

Apply the rule: \((-2, 3) \to (-4 - (-2), 4 - 3) = (-2, 1)\)
New vertex: \(D'' = (-2, 1)\)

Plot \(A''(-1, -2)\), \(B''(1, 1)\), \(C''(0, 2)\), \(D''(-2, 1)\) and connect them to form the rotated quadrilateral.

Final Answer (Vertex Coordinates)

• 90° Clockwise (Origin): \(A'(6, 3)\), \(B'(3, 5)\), \(C'(2, 4)\), \(D'(3, 2)\)
• 180° Around \((-2, 2)\): \(A''(-1, -2)\), \(B''(1, 1)\), \(C''(0, 2)\), \(D''(-2, 1)\)

(Note: Coordinates are approximate based on grid estimation. Adjust if the original quadrilateral has different coordinates.)

Answer:

To solve the rotation problems for quadrilateral \(ABCD\), we first need the coordinates of its vertices. Let's assume the coordinates (from the graph, estimating the grid):

• Let \(A = (-3, 6)\), \(B = (-5, 3)\), \(C = (-4, 2)\), \(D = (-2, 3)\) (adjusted for the grid; actual coordinates may vary slightly, but the method remains the same).

Part (a): Rotate \(90^\circ\) clockwise around the origin

The rule for a \(90^\circ\) clockwise rotation about the origin is:
\((x, y) \to (y, -x)\)

Step 1: Rotate \(A(-3, 6)\)

Apply the rule: \((-3, 6) \to (6, 3)\) (since \(x = -3\), \(y = 6\); new \(x = 6\), new \(y = -(-3) = 3\)).
New vertex: \(A' = (6, 3)\)

Step 2: Rotate \(B(-5, 3)\)

Apply the rule: \((-5, 3) \to (3, 5)\) (new \(x = 3\), new \(y = -(-5) = 5\)).
New vertex: \(B' = (3, 5)\)

Step 3: Rotate \(C(-4, 2)\)

Apply the rule: \((-4, 2) \to (2, 4)\) (new \(x = 2\), new \(y = -(-4) = 4\)).
New vertex: \(C' = (2, 4)\)

Step 4: Rotate \(D(-2, 3)\)

Apply the rule: \((-2, 3) \to (3, 2)\) (new \(x = 3\), new \(y = -(-2) = 2\)).
New vertex: \(D' = (3, 2)\)

Plot \(A'(6, 3)\), \(B'(3, 5)\), \(C'(2, 4)\), \(D'(3, 2)\) and connect them to form the rotated quadrilateral.

Part (b): Rotate \(180^\circ\) around \((-2, 2)\)

The rule for a \(180^\circ\) rotation about a point \((h, k)\) is:
\((x, y) \to (2h - x, 2k - y)\)
Here, \(h = -2\), \(k = 2\), so the rule becomes:
\((x, y) \to (2(-2) - x, 2(2) - y) = (-4 - x, 4 - y)\)

Step 1: Rotate \(A(-3, 6)\)

Apply the rule: \((-3, 6) \to (-4 - (-3), 4 - 6) = (-1, -2)\)
New vertex: \(A'' = (-1, -2)\)

Step 2: Rotate \(B(-5, 3)\)

Apply the rule: \((-5, 3) \to (-4 - (-5), 4 - 3) = (1, 1)\)
New vertex: \(B'' = (1, 1)\)

Step 3: Rotate \(C(-4, 2)\)

Apply the rule: \((-4, 2) \to (-4 - (-4), 4 - 2) = (0, 2)\)
New vertex: \(C'' = (0, 2)\)

Step 4: Rotate \(D(-2, 3)\)

Apply the rule: \((-2, 3) \to (-4 - (-2), 4 - 3) = (-2, 1)\)
New vertex: \(D'' = (-2, 1)\)

Plot \(A''(-1, -2)\), \(B''(1, 1)\), \(C''(0, 2)\), \(D''(-2, 1)\) and connect them to form the rotated quadrilateral.

Final Answer (Vertex Coordinates)

• 90° Clockwise (Origin): \(A'(6, 3)\), \(B'(3, 5)\), \(C'(2, 4)\), \(D'(3, 2)\)
• 180° Around \((-2, 2)\): \(A''(-1, -2)\), \(B''(1, 1)\), \(C''(0, 2)\), \(D''(-2, 1)\)

(Note: Coordinates are approximate based on grid estimation. Adjust if the original quadrilateral has different coordinates.)