Question

3 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. 4 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) 5 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: Solve problem 3

To draw the rotations of Ximena's X - shape:

• For 90 - degree counter - clockwise rotation, apply the rule $(x,y)\to(-y,x)$ to each vertex of the X - shape.
• For 180 - degree clockwise rotation, apply the rule $(x,y)\to(-x,-y)$ to each vertex of the X - shape.
• For 270 - degree counter - clockwise rotation, apply the rule $(x,y)\to(y,-x)$ to each vertex of the X - shape. Then label each image with the degree of rotation and direction.

Step3: Solve problem 4

Let's check the transformation of the vertices of $\triangle ABC$ to $\triangle A'B'C'$.
For point $A(2,3)$ and $A'(3, - 2)$, if we consider the rotation rule. A 90 - degree clockwise rotation about the origin has the rule $(x,y)\to(y,-x)$. For $A(2,3)$, applying $(x,y)\to(y,-x)$ gives $(3,-2)$. For $B(4,4)$ to $B'(4,-4)$ and $C(3,0)$ to $C'(0,-3)$, the rotation is 90 - degree clockwise about the origin.

Step4: Solve problem 5a

The rule for a 90 - degree clockwise rotation about the origin is $(x,y)\to(y,-x)$. If Destiny made an error, it is likely that she did not apply the correct rotation rule. Maybe she used the wrong rotation rule, such as a counter - clockwise rotation rule instead of a clockwise one.

Step5: Solve problem 5b

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

Step6: Solve problem 5c

Using the correct coordinates from part (b), plot the vertices of $\triangle E'F'G'$ in the coordinate plane to draw the correct image.

Answer:

For problem 3: Draw the rotated X - shapes as per the rules and label them.
For problem 4: 90 - degree clockwise rotation.
For problem 5a: Likely used wrong rotation rule.
For problem 5b: Apply $(x,y)\to(y,-x)$ to get correct coordinates.
For problem 5c: Plot the correct - coordinated vertices to draw $\triangle E'F'G'$.