Determine if Triangle ABC with Sides 2X and 12-X is Equilateral

Given the size of the 3 sides of the triangle, is it an equilateral triangle?

To determine if the triangle is equilateral, we need to check if all three sides of the triangle are equal.

The given side lengths are $2X$, $12 - X$, and $12 - X$.

For the triangle to be equilateral, we must have the equality:

• $2X = 12 - X$

Let's solve this equation:

$\begin{aligned} 2X &= 12 - X \\ 2X + X &= 12 \\ 3X &= 12 \\ X &= \frac{12}{3} \\ X &= 4 \end{aligned}$

Substitute $X = 4$ back into the expressions for the sides:

• $2X = 2(4) = 8$

• $12 - X = 12 - 4 = 8$

• The third side, also $12 - X = 8$.

The three calculated side lengths are $8$, $8$, and $8$.

Since all three sides are equal, the triangle is an equilateral triangle.

Therefore, the answer is Yes, the triangle is equilateral.