Question

#1 the coordinates of a point and its image are given below. if it is rotated about the origin. what is the angle of rotation? (6,7)→(7, - 6)
#2 the coordinates of a point and its image are given below. if it is rotated about the origin. what is the angle of rotation? (-2,-3)→(3,-2)
#3 the coordinates of a point and its image are given below. if it is rotated about the origin. what is the angle of rotation? (-1,-8)→(1,8)
#4 the coordinates of a point and its image are given below. if it is rotated about the origin. what is the angle of rotation? (12,-10)→(-10,-12)
#5 rotate the figure below 90° clockwise about the origin.
#6 rotate the figure below 270° clockwise about the origin.
#7 rotate the figure below 180° about the origin.
#8 describe the rotation.

Explanation:

Step1: Recall rotation rules

For a 90 - degree clockwise rotation about the origin, the transformation rule for a point $(x,y)$ is $(x,y)\to(y, - x)$. For a 180 - degree rotation about the origin, the rule is $(x,y)\to(-x,-y)$. For a 270 - degree clockwise (or 90 - degree counter - clockwise) rotation about the origin, the rule is $(x,y)\to(-y,x)$.

Step2: Analyze #1

Given $(6,7)\to(7, - 6)$, comparing with the rules, it is a 90 - degree clockwise rotation.

Step3: Analyze #2

Given $(-2,-3)\to(3, - 2)$, it is a 90 - degree counter - clockwise (270 - degree clockwise) rotation.

Step4: Analyze #3

Given $(-1,-8)\to(1,8)$, it is a 180 - degree rotation.

Step5: Analyze #4

Given $(12,-10)\to(-10,-12)$, it is a 90 - degree counter - clockwise (270 - degree clockwise) rotation.

Step6: Analyze #5

For a 90 - degree clockwise rotation of a figure, use the rule $(x,y)\to(y, - x)$ for each vertex of the rectangle.

Step7: Analyze #6

For a 270 - degree clockwise rotation of a figure, use the rule $(x,y)\to(-y,x)$ for each vertex of the triangle.

Step8: Analyze #7

For a 180 - degree rotation of a figure, use the rule $(x,y)\to(-x,-y)$ for each vertex of the diamond - shaped figure.

Step9: Analyze #8

The rotation from triangle A to triangle B is a 180 - degree rotation as the orientation is reversed and the points follow the $(x,y)\to(-x,-y)$ rule.

Answer:

#1: 90 - degree clockwise
#2: 90 - degree counter - clockwise (270 - degree clockwise)
#3: 180 - degree
#4: 90 - degree counter - clockwise (270 - degree clockwise)
#5: Use $(x,y)\to(y, - x)$ for each vertex of the rectangle
#6: Use $(x,y)\to(-y,x)$ for each vertex of the triangle
#7: Use $(x,y)\to(-x,-y)$ for each vertex of the figure
#8: 180 - degree rotation