Asked by D4VD
Point B is the image of point A when point A is rotated about the origin. What is known about point A and B?
On a coordinate plane, point A is (negative 3, 4) and point B is (4, 3).
Point B is the result of a 180° rotation.
Point A and B have the same x-coordinate.
Point B is the result of a 90° counterclockwise rotation.
Point A and point B are the same distance from the origin.
All Answers 1
Answered by
GPT-5 mini
AI
Only the last statement is true: "Point A and point B are the same distance from the origin."
Reasoning:
- Distance: |A| = sqrt((-3)^2+4^2) = 5, |B| = sqrt(4^2+3^2) = 5, so distances are equal (rotations preserve distance).
- 90° counterclockwise? A 90° CCW rotation sends (x,y) → (−y,x). For A(-3,4) that gives (−4,−3), not B, so that statement is false.
- 180° rotation sends (x,y) → (−x,−y), giving (3,−4), not B, so false.
- Same x-coordinate? A.x = −3, B.x = 4, so false.
(Actually B is the result of a 90° clockwise rotation of A, equivalently a 270° counterclockwise rotation.)
Reasoning:
- Distance: |A| = sqrt((-3)^2+4^2) = 5, |B| = sqrt(4^2+3^2) = 5, so distances are equal (rotations preserve distance).
- 90° counterclockwise? A 90° CCW rotation sends (x,y) → (−y,x). For A(-3,4) that gives (−4,−3), not B, so that statement is false.
- 180° rotation sends (x,y) → (−x,−y), giving (3,−4), not B, so false.
- Same x-coordinate? A.x = −3, B.x = 4, so false.
(Actually B is the result of a 90° clockwise rotation of A, equivalently a 270° counterclockwise rotation.)
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