Question
Triangle ABC has coordinates A(–4, 1), B(–3, 2), and C(–2, 1). The triangle is rotated about the origin and has image coordinates A’(1, 4), B’(2, 3), and C’(1, 2).
What is the angle and direction of rotation?
180° clockwise
180° counterclockwise
90° counterclockwise
90° clockwise
What is the angle and direction of rotation?
180° clockwise
180° counterclockwise
90° counterclockwise
90° clockwise
Answers
GPT-4o mini
To determine the angle and direction of rotation that transforms triangle ABC into triangle A'B'C', we can use the following steps:
1. **Identify the coordinates before and after rotation**:
- A(-4, 1) to A'(1, 4)
- B(-3, 2) to B'(2, 3)
- C(-2, 1) to C'(1, 2)
2. **Translate the coordinates into vectors from the origin**:
- A to the origin (0,0): \((-4, 1)\)
- B to the origin: \((-3, 2)\)
- C to the origin: \((-2, 1)\)
3. **Analyze one point for rotation** (let's start with point A to see the transformation):
- Original point A(-4, 1) and image A'(1, 4).
4. **Calculate the angle of rotation for point A**:
- The angle before rotation (from the positive x-axis to point A) can be found using the arctangent function:
\[
\theta_A = \arctan\left(\frac{1}{-4}\right) \text{(in the second quadrant)}
\]
- The angle after rotation for A'(1, 4):
\[
\theta_{A'} = \arctan\left(\frac{4}{1}\right) \text{(1st quadrant)}
\]
5. **Find the angle difference**:
- The original angle \( \theta_A \) is approximately \( 180^\circ - \frac{\pi}{4} \) (or 180° plus an additional angle in radians).
- The new angle \( \theta_{A'} \) is about \( 90^\circ \).
6. **Calculating final angles**:
- The full angle calculates as a difference between where the angles landed.
We notice that if we compute the transformation in terms of components, we see a half-rotation:
- The transformation does preserve distance because every length remains consistent as it is a uniform rotation.
After testing points B and C similarly, they all adhere to this behavior of movement indicating therefore that the transformation overall is a rotation rather than reflection.
Thus, the net movement through analysis or hard observation indicates that:
The triangle undergoes a **rotation of 90° counterclockwise** to transform from ABC to A'B'C'.
So the answer is:
**90° counterclockwise**.
1. **Identify the coordinates before and after rotation**:
- A(-4, 1) to A'(1, 4)
- B(-3, 2) to B'(2, 3)
- C(-2, 1) to C'(1, 2)
2. **Translate the coordinates into vectors from the origin**:
- A to the origin (0,0): \((-4, 1)\)
- B to the origin: \((-3, 2)\)
- C to the origin: \((-2, 1)\)
3. **Analyze one point for rotation** (let's start with point A to see the transformation):
- Original point A(-4, 1) and image A'(1, 4).
4. **Calculate the angle of rotation for point A**:
- The angle before rotation (from the positive x-axis to point A) can be found using the arctangent function:
\[
\theta_A = \arctan\left(\frac{1}{-4}\right) \text{(in the second quadrant)}
\]
- The angle after rotation for A'(1, 4):
\[
\theta_{A'} = \arctan\left(\frac{4}{1}\right) \text{(1st quadrant)}
\]
5. **Find the angle difference**:
- The original angle \( \theta_A \) is approximately \( 180^\circ - \frac{\pi}{4} \) (or 180° plus an additional angle in radians).
- The new angle \( \theta_{A'} \) is about \( 90^\circ \).
6. **Calculating final angles**:
- The full angle calculates as a difference between where the angles landed.
We notice that if we compute the transformation in terms of components, we see a half-rotation:
- The transformation does preserve distance because every length remains consistent as it is a uniform rotation.
After testing points B and C similarly, they all adhere to this behavior of movement indicating therefore that the transformation overall is a rotation rather than reflection.
Thus, the net movement through analysis or hard observation indicates that:
The triangle undergoes a **rotation of 90° counterclockwise** to transform from ABC to A'B'C'.
So the answer is:
**90° counterclockwise**.