Question
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.
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.
Answers
Answered by
GPT-4o mini
To evaluate the relationship between points A and B given that point A is (-3, 4) and point B is (4, 3), we can analyze each statement:
1. **Point B is the result of a 180° rotation.**
- A 180° rotation around the origin transforms (x, y) to (-x, -y). Rotating point A (-3, 4) by 180° gives (3, -4), which does not match point B (4, 3). Therefore, **this statement is false**.
2. **Point A and B have the same x-coordinate.**
- Point A has an x-coordinate of -3 and point B has an x-coordinate of 4. Since -3 ≠ 4, **this statement is false**.
3. **Point B is the result of a 90° counterclockwise rotation.**
- A 90° counterclockwise rotation transforms (x, y) to (-y, x). Applying this to point A (-3, 4) results in (-4, -3), which does not match point B (4, 3). Therefore, **this statement is false**.
4. **Point A and point B are the same distance from the origin.**
- The distance from the origin (0, 0) to a point (x, y) is calculated using the formula √(x² + y²).
- For point A (-3, 4): Distance = √((-3)² + (4)²) = √(9 + 16) = √25 = 5.
- For point B (4, 3): Distance = √((4)² + (3)²) = √(16 + 9) = √25 = 5.
- Since both distances are equal (5 = 5), **this statement is true**.
In conclusion, out of the options provided, **the only true statement is that point A and point B are the same distance from the origin.**
1. **Point B is the result of a 180° rotation.**
- A 180° rotation around the origin transforms (x, y) to (-x, -y). Rotating point A (-3, 4) by 180° gives (3, -4), which does not match point B (4, 3). Therefore, **this statement is false**.
2. **Point A and B have the same x-coordinate.**
- Point A has an x-coordinate of -3 and point B has an x-coordinate of 4. Since -3 ≠ 4, **this statement is false**.
3. **Point B is the result of a 90° counterclockwise rotation.**
- A 90° counterclockwise rotation transforms (x, y) to (-y, x). Applying this to point A (-3, 4) results in (-4, -3), which does not match point B (4, 3). Therefore, **this statement is false**.
4. **Point A and point B are the same distance from the origin.**
- The distance from the origin (0, 0) to a point (x, y) is calculated using the formula √(x² + y²).
- For point A (-3, 4): Distance = √((-3)² + (4)²) = √(9 + 16) = √25 = 5.
- For point B (4, 3): Distance = √((4)² + (3)²) = √(16 + 9) = √25 = 5.
- Since both distances are equal (5 = 5), **this statement is true**.
In conclusion, out of the options provided, **the only true statement is that point A and point B are the same distance from the origin.**