Question

what is the image of the point (7, - 5) after a rotation of 270° counterclockwise about the origin? (7,-5) what is the image of the point (0,9) after a rotation of 90° counterclockwise about the origin? what is the image of the point (8, - 6) after a rotation of 180° counterclockwise about the origin?

Explanation:

Step1: Recall 180 - degree rotation rule

For a 180 - degree counter - clockwise rotation about the origin, the rule is $(x,y)\to(-x,-y)$. Given the point $(8, - 6)$, applying the rule:
$x = 8$ and $y=-6$, so the new $x$ value is $-8$ and the new $y$ value is $6$. The image is $(-8,6)$.

Step2: Recall 90 - degree rotation rule

For a 90 - degree counter - clockwise rotation about the origin, the rule is $(x,y)\to(-y,x)$. Given the point $(0,9)$, applying the rule:
$x = 0$ and $y = 9$, so the new $x$ value is $-9$ and the new $y$ value is $0$. The image is $(-9,0)$.

Step3: Recall 270 - degree rotation rule

A 270 - degree counter - clockwise rotation about the origin is equivalent to a 90 - degree clockwise rotation. The rule for a 90 - degree clockwise rotation is $(x,y)\to(y,-x)$. Given the point $(7,-5)$, applying the rule:
$x = 7$ and $y=-5$, so the new $x$ value is $-5$ and the new $y$ value is $-7$. The image is $(-5,-7)$.

Answer:

1. $(-8,6)$
2. $(-9,0)$
3. $(-5,-7)$