Triangle GHI is isosceles with GH = HI. The perimeter of triangle GHI is 28 centimeters, and GI = 10...

1. TRANSLATE the problem information

• Given information:
  • Triangle GHI is isosceles with \(\mathrm{GH = HI}\)
  • Perimeter = 28 centimeters
  • \(\mathrm{GI = 10}\) centimeters
  • Find: length of GH
• What this tells us: Two sides are equal, and we know the third side and total perimeter.

2. INFER the approach

• Since \(\mathrm{GH = HI}\), we can use one variable (let's call it x) to represent both equal sides
• The perimeter equation will be: \(\mathrm{x + x + 10 = 28}\)
• This gives us a simple linear equation to solve

3. TRANSLATE into mathematical equation

• Let \(\mathrm{GH = HI = x}\)
• Perimeter equation: \(\mathrm{x + x + 10 = 28}\)
• Simplified: \(\mathrm{2x + 10 = 28}\)

4. SIMPLIFY through algebraic steps

• \(\mathrm{2x + 10 = 28}\)
• Subtract 10 from both sides: \(\mathrm{2x = 18}\)
• Divide by 2: \(\mathrm{x = 9}\)
• Therefore: \(\mathrm{GH = 9}\) centimeters

Answer: C) 9

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't properly identify which sides are equal in the isosceles triangle. They might assume all three sides are equal (confusing isosceles with equilateral) or incorrectly identify which two sides are equal.

If they think all sides are equal: \(\mathrm{3x = 28}\), so \(\mathrm{x ≈ 9.33}\), leading them to select Choice C (9) as the closest answer, but through wrong reasoning.

If they think \(\mathrm{GH = GI = 10}\): Then \(\mathrm{HI = 28 - 10 - 10 = 8}\), leading them to select Choice B (8).

Second Most Common Error:

Poor TRANSLATE reasoning: Students set up the wrong equation by misunderstanding the perimeter concept or the given information.

They might write \(\mathrm{x + 10 + 10 = 28}\) (assuming GI appears twice somehow), leading to \(\mathrm{x = 8}\) and selecting Choice B (8).

The Bottom Line:

This problem tests whether students can correctly interpret "isosceles with \(\mathrm{GH = HI}\)" and translate that into a proper algebraic setup. The key insight is recognizing that exactly two sides are equal, then using that constraint to create a solvable equation.