Equilateral Triangle Verification: Comparing Sides 10+X, 5X-10, and 3X

To determine if the given triangle is equilateral, we need to ensure that all sides are equal. Given the sides:

• $a = 10 + X$
• $b = 10 + X$
• $c = 5X - 10$

Since two sides are already equal ($a = b$), we need to check if this is equal to the third side:

Set the side lengths $a$ and $c$ equal:

$10 + X = 5X - 10$

Solve for $X$:

$10 + X = 5X - 10$

Subtract $X$ from both sides:

$10 = 4X - 10$

Add 10 to both sides:

$20 = 4X$

Divide by 4:

$X = 5$

Substitute $X = 5$ back into the side length expressions:

• $10 + X = 10 + 5 = 15$
• $5X - 10 = 5(5) - 10 = 25 - 10 = 15$

All sides are equal to 15. Hence, the triangle is equilateral.

Therefore, the solution to the problem is Yes.