https://file.aiquickdraw.com/mathbot/file/172970333281123g0kpw9.jpg

To solve this problem, we note that \( \triangle ABC \) is equilateral, meaning all sides are equal.

The side lengths are given as:
- \( AB = 10x - 6 \)
- \( BC = 7x + 3 \)
- \( AC = 8x \)

Since all sides of an equilateral triangle are equal, we set the expressions for the sides equal to each other:

1. \( AB = BC \)
\[
10x - 6 = 7x + 3
\]
Solving for \( x \):
\[
10x - 7x = 3 + 6
\]
\[
3x = 9
\]
\[
x = 3
\]

2. Now substitute \( x = 3 \) into each side expression to find their lengths:
- \( AB = 10(3) - 6 = 30 - 6 = 24 \)
- \( BC = 7(3) + 3 = 21 + 3 = 24 \)
- \( AC = 8(3) = 24 \)

Thus, \( x = 3 \), and the side lengths are \( AB = BC = AC = 24 \).

The correct answer is:
\[
\boxed{x = 3, AB = 24, BC = 24, AC = 24}
\]

Would you like further details or clarifications? 

Here are 5 additional questions related to this problem:
1. What are the properties of an equilateral triangle that help in solving this type of problem?
2. How do you verify if a triangle is equilateral using algebraic expressions?
3. How can you use systems of equations to solve for multiple unknowns in geometric problems?
4. What happens to the interior angles of an equilateral triangle, and why?
5. Can you apply this method to solve for the sides of an isosceles triangle?

**Tip:** In any geometric problem involving an equilateral triangle, always start by setting all sides equal to one another, as this simplifies solving for unknowns.

Find x, AB, BC and AC if △ABC is equilateral. Given: AB = 10x - 6, BC = 7x + 3, AC = 8x

Learn how to solve for x and the side lengths of an equilateral triangle △ABC using algebraic expressions for AB, BC, and AC. This problem involves setting up and solving equations for an equilateral triangle, a key concept in geometry and algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation