The perimeter of a triangle is given by the expression 14x + 8. The lengths of two of the sides...
GMAT Advanced Math : (Adv_Math) Questions
The perimeter of a triangle is given by the expression \(14\mathrm{x} + 8\). The lengths of two of the sides of the triangle are \(3\mathrm{x} + 2\) and \(5\mathrm{x} - 1\). Which of the following expressions represents the length of the third side of the triangle?
1. TRANSLATE the problem information
- Given information:
- Total perimeter: \(14\mathrm{x} + 8\)
- Known side 1: \(3\mathrm{x} + 2\)
- Known side 2: \(5\mathrm{x} - 1\)
- Unknown side 3: ?
- What this tells us: Since perimeter = sum of all sides, we can write an equation
2. INFER the approach
- We need to use the fact that: Side 1 + Side 2 + Side 3 = Perimeter
- Strategy: Find the sum of the two known sides, then subtract from total perimeter
3. SIMPLIFY to find the sum of known sides
- Add the two known sides: \((3\mathrm{x} + 2) + (5\mathrm{x} - 1)\)
- Combine like terms: \(3\mathrm{x} + 5\mathrm{x} + 2 + (-1) = 8\mathrm{x} + 1\)
4. SIMPLIFY to solve for the third side
- Set up equation: \((8\mathrm{x} + 1) + \text{Side 3} = 14\mathrm{x} + 8\)
- Isolate Side 3: \(\text{Side 3} = (14\mathrm{x} + 8) - (8\mathrm{x} + 1)\)
- Distribute the negative sign: \(\text{Side 3} = 14\mathrm{x} + 8 - 8\mathrm{x} - 1\)
- Combine like terms: \(\text{Side 3} = (14\mathrm{x} - 8\mathrm{x}) + (8 - 1) = 6\mathrm{x} + 7\)
Answer: A (\(6\mathrm{x} + 7\))
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Students make sign errors when distributing the negative sign in the subtraction step.
Instead of correctly computing \((14\mathrm{x} + 8) - (8\mathrm{x} + 1) = 14\mathrm{x} + 8 - 8\mathrm{x} - 1\), they might forget to distribute the negative to both terms, calculating \((14\mathrm{x} + 8) - (8\mathrm{x} + 1) = 14\mathrm{x} + 8 - 8\mathrm{x} + 1 = 6\mathrm{x} + 9\).
This leads them to select Choice B (\(6\mathrm{x} + 9\)).
Second Most Common Error:
Poor INFER reasoning: Students misunderstand the relationship and think they should add all the given expressions together instead of using subtraction to find the missing piece.
They incorrectly compute: \((14\mathrm{x} + 8) + (3\mathrm{x} + 2) + (5\mathrm{x} - 1) = 22\mathrm{x} + 9\), thinking this represents the third side.
This leads them to select Choice C (\(22\mathrm{x} + 9\)).
The Bottom Line:
This problem requires careful attention to both the conceptual relationship (perimeter as sum of sides) and precise algebraic manipulation, especially with sign distribution. Students who rush through the algebra or misunderstand the setup are most likely to make errors.