Question

ximena draws an x shape in the coordinate plane as shown. draw the images of ximenas x shape rotated 90° counterclockwise, 180° clockwise, and 270° counterclockwise around the origin. label each image with the degree of rotation and direction.
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)
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 clockwise (or counter - clockwise) rotation about the origin, the rule is $(x,y)\to(-x,-y)$. For a 270 - degree counter - clockwise rotation about the origin, the rule is $(x,y)\to(y, - x)$.

Step2: Analyze question 4

Given $\triangle ABC$ with $A(2,3)$, $B(4,4)$, $C(3,0)$ and $\triangle A'B'C'$ with $A'(3, - 2)$, $B'(4, - 4)$, $C'(0, - 3)$. If we apply the 90 - degree clockwise rotation rule $(x,y)\to(y,-x)$ to the points of $\triangle ABC$, for $A(2,3)$ we get $(3,-2)$, for $B(4,4)$ we get $(4,-4)$ and for $C(3,0)$ we get $(0,-3)$. So the rotation is 90 degrees clockwise.

Step3: Analyze question 5a

The error Destiny likely made is using the wrong rotation rule. For a 90 - degree clockwise rotation about the origin, the rule is $(x,y)\to(y,-x)$, and she might have used a different (incorrect) transformation rule.

Step4: Analyze question 5b

To find the correct coordinates for a 90 - degree clockwise rotation of $\triangle EFG$ about the origin, use the rule $(x,y)\to(y,-x)$ for each vertex of $\triangle EFG$.

Step5: Analyze question 5c

To draw $\triangle E'F'G'$, plot the correct vertices (found in step 4) on the coordinate plane.

Answer:

For question 4: The rotation is 90 degrees clockwise.
For question 5a: Likely used the wrong rotation rule.
For question 5b: Use the rule $(x,y)\to(y,-x)$ for each vertex of $\triangle EFG$ to find the correct coordinates.
For question 5c: Plot the vertices found in 5b on the coordinate plane to draw $\triangle E'F'G'$.