**Finding Interior Angles of a Triangle using Vectors****Problem Statement:**Use vectors to find the interior angles of the triangle with the given vertices. (Round your answers to two decimal places)Vertices:1. (-7, -8)2. (5, 9)3. (5, 3)**Tasks:**1. Calculate the smallest interior angle.2. Calculate the largest interior angle.**Response Fields:**1. Smallest angle: ____° (smallest value)2. Second angle: ____°3. Largest angle: ____° (largest value)To find the interior angles of a triangle using vectors, follow these steps:1. **Define Vectors:** - Identify the vertices of the triangle as points in a coordinate system. - For the points A(-7, -8), B(5, 9), and C(5, 3), define the vectors connecting these points.2. **Calculate Vectors:** - Vector AB: From A to B - Vector BC: From B to C - Vector CA: From C to A3. **Dot Product and Magnitudes:** - Calculate the dot products of each pair of vectors. - Find the magnitudes of the vectors.4. **Cosine of Angles:** - Use the dot product formula to find the cosines of the angles. - Dot product formula: $ \vec{u} \cdot \vec{v} = |\vec{u}| |\vec{v}| \cos \theta $5. **Angles:** - Calculate the angles using the inverse cosine function: $ \theta = \cos^{-1} \left( \frac{\vec{u} \cdot \vec{v}}{|\vec{u}| |\vec{v}|} \right) $6. **Convert to Degrees:** - Ensure the angles are in degrees (if using a calculator, ensure it is in degree mode).7. **Order the Angles:** - Determine which angle is the smallest and largest.Use this method to find the precise values of the angles at each vertex of the triangle formed by the given vertices.

Answered: Image /qna-images/answer/d63144a9-dd34-4177-bad0-4e2cefe1d69b.jpg

Answered: Use vectors to find the interior angles…

Use vectors to find the interior angles of the triangle with the given vertices. (Round your answers to t (-7, -8), (5, 9), (5, 3) (smallest value) ° (largest value)

Holt Mcdougal Larson Pre-algebra: Student Edition 2012

1st Edition

ISBN:9780547587776

Author:HOLT MCDOUGAL

Publisher:HOLT MCDOUGAL

Chapter12: Angle Relationships And Transformations

Section12.5: Reflections And Symmetry

Problem 20E

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Question

**Finding Interior Angles of a Triangle using Vectors**

**Problem Statement:**

Use vectors to find the interior angles of the triangle with the given vertices. (Round your answers to two decimal places)

Vertices:
1. (-7, -8)
2. (5, 9)
3. (5, 3)

**Tasks:**

1. Calculate the smallest interior angle.
2. Calculate the largest interior angle.

**Response Fields:**

1. Smallest angle: ____° (smallest value)
2. Second angle: ____°
3. Largest angle: ____° (largest value)

To find the interior angles of a triangle using vectors, follow these steps:

1. **Define Vectors:**
- Identify the vertices of the triangle as points in a coordinate system.
- For the points A(-7, -8), B(5, 9), and C(5, 3), define the vectors connecting these points.

2. **Calculate Vectors:**
- Vector AB: From A to B
- Vector BC: From B to C
- Vector CA: From C to A

3. **Dot Product and Magnitudes:**
- Calculate the dot products of each pair of vectors.
- Find the magnitudes of the vectors.

4. **Cosine of Angles:**
- Use the dot product formula to find the cosines of the angles.
- Dot product formula: $ \vec{u} \cdot \vec{v} = |\vec{u}| |\vec{v}| \cos \theta $

5. **Angles:**
- Calculate the angles using the inverse cosine function: $ \theta = \cos^{-1} \left( \frac{\vec{u} \cdot \vec{v}}{|\vec{u}| |\vec{v}|} \right) $

6. **Convert to Degrees:**
- Ensure the angles are in degrees (if using a calculator, ensure it is in degree mode).

7. **Order the Angles:**
- Determine which angle is the smallest and largest.

Use this method to find the precise values of the angles at each vertex of the triangle formed by the given vertices.

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