Question

△abc is the image of △abc under a rotation about the origin, (0,0).
graph of coordinate plane with grid, showing △abc (blue) and △abc (red).
determine the angles of rotation.
choose all answers that apply:
a 90° clockwise
b 90° counterclockwise
c 180°
d 270° clockwise
e 270° counterclockwise

Explanation:

Step1: Identify original points

Original coordinates: $A(-4, 2)$, $B(-5, -3)$, $C(-3, 1)$

Step2: Identify image points

Image coordinates: $A'(-2, -4)$, $B'(3, -5)$, $C'(-1, -3)$

Step3: Test 90° clockwise rule

Rule: $(x,y)\to(y,-x)$
For $A(-4,2)$: $(2, 4)$ ≠ $A'(-2,-4)$ → Not valid.

Step4: Test 90° counterclockwise rule

Rule: $(x,y)\to(-y,x)$
For $A(-4,2)$: $(-2, -4)$ = $A'(-2,-4)$;
For $B(-5,-3)$: $(3, -5)$ = $B'(3,-5)$;
For $C(-3,1)$: $(-1, -3)$ = $C'(-1,-3)$ → Valid.

Step5: Equivalent rotation check

$90^\circ$ counterclockwise = $270^\circ$ clockwise.
Verify $270^\circ$ clockwise rule: $(x,y)\to(-y,x)$, same as above → Valid.

Step6: Test 180° rule

Rule: $(x,y)\to(-x,-y)$
For $A(-4,2)$: $(4, -2)$ ≠ $A'(-2,-4)$ → Not valid.

Step7: Test 270° counterclockwise rule

Rule: $(x,y)\to(y,-x)$
For $A(-4,2)$: $(2, 4)$ ≠ $A'(-2,-4)$ → Not valid.

Answer:

B. $90^\circ$ counterclockwise
D. $270^\circ$ clockwise