The lengths of two sides of a triangle are 4 centimeters and 6 centimeters. If the perimeter of the triangle...

1. TRANSLATE the problem information

• Given information:
  • Two sides of triangle: \(\mathrm{4\text{ cm}}\) and \(\mathrm{6\text{ cm}}\)
  • Perimeter of triangle: \(\mathrm{18\text{ cm}}\)
  • Need to find: third side length

• Let \(\mathrm{x}\) = length of third side (in cm)

2. TRANSLATE the perimeter relationship

• Perimeter means the sum of all three side lengths
• This gives us the equation: \(\mathrm{4 + 6 + x = 18}\)

3. SIMPLIFY to solve for x

• Combine the known sides: \(\mathrm{10 + x = 18}\)
• Subtract 10 from both sides: \(\mathrm{x = 18 - 10}\)
• Therefore: \(\mathrm{x = 8}\)

Answer: B. 8

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when solving the equation \(\mathrm{10 + x = 18}\)

Some students might incorrectly calculate \(\mathrm{18 - 10}\) or confuse the order of subtraction. For example, they might compute \(\mathrm{10 - 8 = 2}\) instead of \(\mathrm{18 - 10 = 8}\).

This may lead them to select Choice A (2).

Second Most Common Error:

Poor TRANSLATE reasoning: Students misunderstand what "third side" means in relation to perimeter

Some students think the third side should equal the sum of the two given sides (\(\mathrm{4 + 6 = 10}\)), not understanding that perimeter involves adding ALL three sides together.

This may lead them to select Choice C (10).

The Bottom Line:

This problem tests whether students truly understand that perimeter means "add up ALL the sides" and can set up and solve a simple linear equation. The arithmetic is straightforward, but the setup requires clear thinking about what perimeter actually means.