Triangle Side Analysis: Comparing 8X, 10X, and 9X Side Lengths

Given the values of the sides of the triangle below, is it a triangle with different sides?

To determine if the given triangle is scalene, we must examine the sides: $8X$, $10X$, and $7X + 2X$.

• Step 1: Simplify the expression for the third side:
  $7X + 2X = 9X$.
  Thus, the sides of the triangle are $8X$, $10X$, and $9X$.
• Step 2: Compare the side lengths:
  - The first side is $8X$.
  - The second side is $10X$.
  - The third side is $9X$.
• Step 3: Check for distinctness of each side:
  Since $8X \neq 9X$, $9X \neq 10X$, and $8X \neq 10X$, all sides are indeed different.

Therefore, the triangle is scalene, as all sides have different lengths.

The solution to the problem is: Yes