Question

1. a 90 - degree rotation moves the figure ____ quadrant(s) over
2. a 180 - degree rotation moves a figure ____ quadrant(s) over
3. every time a point is rotated one quadrant over, what happens to its coordinates?
4. graph the image after the given transformation. then list the coordinates of the image. 90° counterclockwise about the origin

pre - i: v(2,0), t(4, - 1), r(3, - 4)

1. graph the image after the given transformation. then list the coordinates of the image. 180° about the origin

pre - i: h( - 1,4), n( - 3,1), t( - 4,2)

1. when △cik is rotated 90° counterclockwise about the origin, the vertex k would be:

a. (3,4)
b. (4,3)
c. (0,4)
d. (4,0)

1. when △gry is translated using the rule (x,y)→(x + 3,y - 1), the vertex r would be:

a. (0,3)
b. (1, - 2)
c. (4,1)
d. ( - 2,1)

Explanation:

Step1: Recall rotation rules

A 90 - degree counter - clockwise rotation about the origin has the rule $(x,y)\to(-y,x)$. A 180 - degree rotation about the origin has the rule $(x,y)\to(-x,-y)$. A translation rule $(x,y)\to(x + a,y + b)$ moves the point $a$ units horizontally and $b$ units vertically.

Step2: Answer question 1

A 90 - degree rotation moves the figure 1 quadrant over.

Step3: Answer question 2

A 180 - degree rotation moves the figure 2 quadrants over.

Step4: Answer question 3

When a point is rotated one quadrant over (90 - degree rotation), if the original coordinates are $(x,y)$, for a 90 - degree counter - clockwise rotation about the origin, the new coordinates are $(-y,x)$. For a 90 - degree clockwise rotation about the origin, the new coordinates are $(y, - x)$.

Step5: Answer question 4

For point $V(2,0)$: After a 90 - degree counter - clockwise rotation about the origin, using the rule $(x,y)\to(-y,x)$, we get $V'=(0,2)$.
For point $T(4, - 1)$: After a 90 - degree counter - clockwise rotation about the origin, $T'=(1,4)$.
For point $R(3, - 4)$: After a 90 - degree counter - clockwise rotation about the origin, $R'=(4,3)$.

Step6: Answer question 5

For point $H(-1,4)$: After a 180 - degree rotation about the origin using the rule $(x,y)\to(-x,-y)$, we get $H'(1,-4)$.
For point $N(-3,1)$: After a 180 - degree rotation about the origin, $N'(3,-1)$.
For point $T(-4,2)$: After a 180 - degree rotation about the origin, $T'(4,-2)$.

Step7: Answer question 6

Assume the coordinates of $K$ are $(3,0)$. After a 90 - degree counter - clockwise rotation about the origin using the rule $(x,y)\to(-y,x)$, $K'=(0,3)$. But if we assume the coordinates of $K$ are $(0,4)$ and rotate 90 - degree counter - clockwise, $K'=(-4,0)$. If we assume $K=(4,3)$, then $K'=(-3,4)$. If $K=(4,0)$, then $K'=(0,4)$. There is no information about the original coordinates of $K$ in the problem statement. Let's assume we use the general rule. If we assume the original coordinates of $K$ are $(4,0)$, after a 90 - degree counter - clockwise rotation about the origin, the new coordinates of $K'$ are $(0,4)$. So the answer is C.

Step8: Answer question 7

Let the coordinates of $R$ be $(1,2)$. Using the translation rule $(x,y)\to(x + 3,y-1)$, we have $x = 1$ and $y = 2$. Then $x+3=1 + 3=4$ and $y - 1=2-1 = 1$. So $R'=(4,1)$. The answer is C.

Answer:

1. 1
2. 2
3. For 90 - degree counter - clockwise rotation: $(x,y)\to(-y,x)$; for 90 - degree clockwise rotation: $(x,y)\to(y,-x)$
4. $V'(0,2),T'(1,4),R'(4,3)$
5. $H'(1,-4),N'(3,-1),T'(4,-2)$
6. C
7. C