Question

draw the image of quadrilateral abcd after each rotation using b as center. a. 90 degrees clockwise b. 120 degrees clockwise c. 30 degrees counterclockwise (from unit 1, lesson 2)

Explanation:

Step1: Recall rotation rules

For a rotation of a point $(x,y)$ about a center $(a,b)$ by an angle $\theta$, we use transformation rules. When rotating a shape, we rotate each vertex of the quadrilateral ABCD about point B.

Step2: Rotate for 90 - degrees clockwise

To rotate a point $(x,y)$ 90 - degrees clockwise about a center $(a,b)$, the new coordinates $(x',y')$ are given by: $x'=a+(y - b)$ and $y'=b-(x - a)$. Apply this to each vertex of quadrilateral ABCD with center of rotation B.

Step3: Rotate for 120 - degrees counter - clockwise

To rotate a point $(x,y)$ counter - clockwise about a center $(a,b)$ by an angle $\theta$, we can use the rotation matrix

$$\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}$$

on the vector

$$\begin{bmatrix}x - a\\y - b\end{bmatrix}$$

and then add the coordinates of the center back. For $\theta = 120^{\circ}$, $\cos120^{\circ}=-\frac{1}{2}$ and $\sin120^{\circ}=\frac{\sqrt{3}}{2}$. Apply to each vertex of ABCD.

Step4: Rotate for 30 - degrees counter - clockwise

Similar to step 3, for $\theta = 30^{\circ}$, $\cos30^{\circ}=\frac{\sqrt{3}}{2}$ and $\sin30^{\circ}=\frac{1}{2}$. Use the rotation matrix

$$\begin{bmatrix}\cos30^{\circ}&-\sin30^{\circ}\\\sin30^{\circ}&\cos30^{\circ}\end{bmatrix}$$

on the vectors from the center B to each vertex of ABCD and then add the coordinates of B back.

Answer:

Drawings of the rotated quadrilaterals according to the above - described rotation rules for each case (a 90 - degrees clockwise, b 120 - degrees counter - clockwise, c 30 - degrees counter - clockwise) with B as the center of rotation.