The perimeter of an isosceles triangle is 83 inches. Each of the two congruent sides of the triangle has a...

1. TRANSLATE the problem information

• Given information:
  • Perimeter = 83 inches
  • Two congruent sides = 24 inches each
  • Unknown third side = x inches

• Set up the equation: \(24 + 24 + \mathrm{x} = 83\)

2. SIMPLIFY to solve for the unknown side

• Combine the known sides: \(24 + 24 = 48\)
• Rewrite equation: \(48 + \mathrm{x} = 83\)
• Subtract 48 from both sides: \(\mathrm{x} = 83 - 48 = 35\)

Answer: 35

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not understand what "congruent sides" means in the context of an isosceles triangle, leading them to set up incorrect equations like \(24 + \mathrm{x} + \mathrm{x} = 83\) (thinking the third side equals one of the congruent sides) or \(83 - 24 = \mathrm{x}\) (forgetting about the second congruent side). This leads to confusion and incorrect calculations.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(24 + 24 + \mathrm{x} = 83\) but make arithmetic errors, such as incorrectly calculating \(24 + 24 = 46\) instead of 48, or making subtraction errors like \(83 - 48 = 33\) instead of 35. These calculation mistakes lead to wrong final answers.

The Bottom Line:

This problem tests whether students can correctly interpret the language of geometry (congruent sides in isosceles triangles) and execute basic algebraic manipulations accurately. Success depends on careful reading and precise arithmetic.