In an isosceles triangle, the two equal sides have the same length, and the third side is called the base....

1. TRANSLATE the problem information

• Given information:
  • Isosceles triangle (two sides are equal)
  • Base = 8 centimeters
  • Perimeter = 26 centimeters
  • Need to find: length of one equal side

• What this tells us: We have two unknown equal sides and one known side (the base)

2. TRANSLATE the setup into mathematics

• Let \(\mathrm{x}\) = length of one equal side
• Since it's isosceles: both equal sides have length \(\mathrm{x}\)
• Perimeter equation: \(\mathrm{x + x + 8 = 26}\)
• This simplifies to: \(\mathrm{2x + 8 = 26}\)

3. SIMPLIFY the equation to find x

• Start with: \(\mathrm{2x + 8 = 26}\)
• Subtract 8 from both sides: \(\mathrm{2x = 18}\)
• Divide both sides by 2: \(\mathrm{x = 9}\)

4. APPLY CONSTRAINTS to verify our answer

• Check triangle inequality (any two sides > third side):
  • \(\mathrm{9 + 9 \gt 8}\)? Yes: \(\mathrm{18 \gt 8}\) ✓
  • \(\mathrm{9 + 8 \gt 9}\)? Yes: \(\mathrm{17 \gt 9}\) ✓
  • \(\mathrm{8 + 9 \gt 9}\)? Yes: \(\mathrm{17 \gt 9}\) ✓

Answer: C (9)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misunderstand what "isosceles triangle" means or incorrectly set up the perimeter equation.

Some students might think all three sides are equal (confusing with equilateral triangle) and set up: \(\mathrm{3x = 26}\), leading to \(\mathrm{x = 8.67}\), which isn't an answer choice. Or they might forget that there are TWO equal sides and set up: \(\mathrm{x + 8 = 26}\), getting \(\mathrm{x = 18}\), which also isn't an answer choice. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{2x + 8 = 26}\) but make arithmetic errors.

A common mistake is:

\(\mathrm{2x + 8 = 26}\)
\(\mathrm{2x = 26 + 8 = 34}\)
\(\mathrm{x = 17}\)

Since 17 isn't among the choices, they might guess or pick the closest option. Another error:

\(\mathrm{2x + 8 = 26}\)
\(\mathrm{x = 26 - 8 = 18}\)
\(\mathrm{x = 9}\)

(accidentally getting the right answer through wrong method, or \(\mathrm{x = 18}\) which isn't a choice).

The Bottom Line:

This problem tests whether students can correctly interpret "isosceles triangle" and translate that understanding into a proper perimeter equation. The algebra itself is straightforward, but the setup requires careful reading and geometric knowledge.