Question

1. kimera draws an x - shape in the coordinate plane as shown. draw the images of kimeras x - shape rotated 90° counterclockwise, 180° clockwise, and 270° counterclockwise around the origin. label each image with the degree of rotation and direction.
2. michael rotates △abc to form the image △abc. the table shows the corresponding vertices for △abc and △abc. what degree of rotation and direction did michael rotate △abc to form △abc?

△abc a(2,3) b(4,4) c(3,0)
△abc a(3, - 2) b(4, - 4) c(0, - 3)

1. destiny plots △efg in the coordinate plane. then destinys teacher asks her to rotate △efg 90° clockwise around the origin to form its image △efg. her original figure and image are shown in the coordinate plane.

a. what error did destiny make?
b. what are the correct coordinates of the vertices for the image?
c. draw the correct image △efg in the coordinate plane.

Explanation:

Step1: Recall rotation rules

For a 90 - degree counter - clockwise rotation about the origin, the rule is $(x,y)\to(-y,x)$. For a 180 - degree rotation about the origin, the rule is $(x,y)\to(-x,-y)$. For a 90 - degree clockwise rotation about the origin, the rule is $(x,y)\to(y, - x)$.

Step2: Analyze Michael's rotation

For point $A(2,3)$ to $A'(3, - 2)$, using the rotation rules, we can see that it is a 90 - degree clockwise rotation. For point $B(4,4)$ to $B'(4, - 4)$ and $C(3,0)$ to $C'(0, - 3)$, all follow the 90 - degree clockwise rotation rule $(x,y)\to(y, - x)$. So Michael rotated $\triangle ABC$ 90 degrees clockwise.

Step3: Analyze Destiny's error

The rule for a 90 - degree clockwise rotation about the origin is $(x,y)\to(y, - x)$. Looking at the graph, it seems Destiny may have used the wrong rotation rule, perhaps confusing it with a counter - clockwise rotation or another transformation.

Step4: Find correct coordinates for Destiny's rotation

Let the vertices of $\triangle EFG$ be $E(x_1,y_1)$, $F(x_2,y_2)$, $G(x_3,y_3)$. Using the 90 - degree clockwise rotation rule $(x,y)\to(y, - x)$, we can find the new coordinates.

Step5: Drawing the image

To draw the correct $\triangle E'F'G'$, plot the new coordinates found in Step 4 on the coordinate plane.

Answer:

1. 90 degrees clockwise
2. a. Likely used wrong rotation rule.

b. Use $(x,y)\to(y, - x)$ for each vertex to find correct coordinates.
c. Plot the correct coordinates from part b on the coordinate plane.