Question

performing a rotation in the coordinate plane

fill in the missing numbers to make each rotation true.

figure wxyz is rotated 90° clockwise around the origin to form figure wxyz.
① w(5, 4)
② x(4, 1)
③ y(1, 6)
④ z(3, 8)
figure wxyz is rotated 180° clockwise around the origin to form figure wxyz.
⑤ w(4, )
⑥ x(1, )
⑦ y(, -1)
⑧ z(, )
figure wxyz is rotated 90° counter - clockwise around the origin to form figure wxyz.
⑨ w(-5, )
⑩ x(, -1)
⑪ y(, )
⑫ z(, )

Explanation:

Step1: Recall 90 - degree clockwise rotation rule

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

Step2: Apply rule to point W

If $W(x,y)$ and $W'(5,y')$, assume the original coordinates of $W$ are $(- 4,5)$. After a 90 - degree clockwise rotation, $W'$ is $(5,4)$.

Step3: Apply rule to point X

If $X(x,y)$ and $X'(4,y')$, assume the original coordinates of $X$ are $(-1,4)$. After a 90 - degree clockwise rotation, $X'$ is $(4,1)$.

Step4: Apply rule to point Y

If $Y(x,y)$ and $Y'(x',6)$, assume the original coordinates of $Y$ are $(-6, - 1)$. After a 90 - degree clockwise rotation, $Y'$ is $( - 1,6)$.

Step5: Apply rule to point Z

If $Z(x,y)$ and $Z'(x',y')$, assume the original coordinates of $Z$ are $(-8, - 3)$. After a 90 - degree clockwise rotation, $Z'$ is $( - 3,8)$.

Step6: Recall 180 - degree clockwise rotation rule

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

Step7: Apply 180 - degree rule to point W

If the original coordinates of $W$ are $(-4, - 5)$, after a 180 - degree clockwise rotation, $W'$ is $(4,5)$.

Step8: Apply 180 - degree rule to point X

If the original coordinates of $X$ are $(-1, - 4)$, after a 180 - degree clockwise rotation, $X'$ is $(1,4)$.

Step9: Apply 180 - degree rule to point Y

If the original coordinates of $Y$ are $(-6, - 1)$, after a 180 - degree clockwise rotation, $Y'$ is $(6,1)$.

Step10: Apply 180 - degree rule to point Z

If the original coordinates of $Z$ are $(-8, - 3)$, after a 180 - degree clockwise rotation, $Z'$ is $(8,3)$.

Step11: Recall 90 - degree counter - clockwise rotation rule

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

Step12: Apply 90 - degree counter - clockwise rule to point W

If the original coordinates of $W$ are $(-4,5)$, after a 90 - degree counter - clockwise rotation, $W'$ is $(-5, - 4)$.

Step13: Apply 90 - degree counter - clockwise rule to point X

If the original coordinates of $X$ are $(-1,4)$, after a 90 - degree counter - clockwise rotation, $X'$ is $(-4,-1)$.

Step14: Apply 90 - degree counter - clockwise rule to point Y

If the original coordinates of $Y$ are $(-6, - 1)$, after a 90 - degree counter - clockwise rotation, $Y'$ is $(1,-6)$.

Step15: Apply 90 - degree counter - clockwise rule to point Z

If the original coordinates of $Z$ are $(-8, - 3)$, after a 90 - degree counter - clockwise rotation, $Z'$ is $(3,-8)$.

Answer:

1. $4$
2. $1$
3. $-1$
4. $-3,8$
5. $5$
6. $4$
7. $6$
8. $8,3$
9. $-4$
10. $-4$
11. $1,-6$
12. $3,-8$