https://file.aiquickdraw.com/mathbot/file/1726565093330uyz6e0sz.jpg

We are tasked with finding the angles of triangle ABC, given the vertices \( A(-7, 4) \), \( B(-3, -2) \), and \( C(-1, 0) \). Let's go step by step to determine the angle measures and deduce information about the triangle.

### Steps to Solve:
1. **Find the lengths of the sides** using the distance formula:
   \[
   d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
   \]
   This will give us the lengths of sides \( AB \), \( BC \), and \( AC \).

2. **Use the Law of Cosines** to find the angles:
   \[
   \cos(\theta) = \frac{b^2 + c^2 - a^2}{2bc}
   \]
   where \( a \), \( b \), and \( c \) are the side lengths.

3. **Sum of the angles** in a triangle is 180°, so we can use this property to check our results.

Let me first calculate the side lengths and then proceed with determining the angles.The side lengths of triangle ABC are:

- \( AB = 7.21 \) units
- \( BC = 2.83 \) units
- \( AC = 7.21 \) units

The measures of the angles are:

- \( \angle A = 22.62^\circ \)
- \( \angle B = 78.69^\circ \)
- \( \angle C = 78.69^\circ \)

### Deduction:
Since two angles of the triangle (B and C) are equal, triangle ABC is **isosceles**.

Would you like further details or explanations?

Here are 5 related questions to consider:
1. How would the process change if one vertex was in 3D space?
2. What is the significance of the Law of Cosines in non-right triangles?
3. Can you use trigonometry to solve a triangle without knowing side lengths?
4. How can you verify that a triangle is isosceles given only its vertices?
5. What properties differentiate an isosceles triangle from a scalene triangle?

**Tip:** When working with isosceles triangles, knowing two equal sides or angles simplifies solving for other properties!

Find the measure of all angles of the triangle ABC with vertices A(-7, 4), B(-3, -2), and C(-1, 0). What can you deduce about the triangle?

Learn how to calculate the angles of triangle ABC using the distance formula and the Law of Cosines. Given the vertices at A(-7, 4), B(-3, -2), and C(-1, 0), we deduce that the triangle is isosceles with equal angles at B and C.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation