In triangle PQR, PR = RQ = 8 centimeters. Which of the following values of the perimeter of triangle PQR...
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
In triangle \(\mathrm{PQR}\), \(\mathrm{PR = RQ = 8}\) centimeters. Which of the following values of the perimeter of triangle \(\mathrm{PQR}\) is sufficient to prove that triangle \(\mathrm{PQR}\) is equilateral?
1. TRANSLATE the problem information
- Given information:
- Triangle PQR with \(\mathrm{PR = RQ = 8}\) centimeters
- Need to find which perimeter proves the triangle is equilateral
- What this tells us: We start with an isosceles triangle (two equal sides)
2. INFER the key relationship
- For a triangle to be equilateral, ALL THREE sides must be equal
- Since \(\mathrm{PR = RQ = 8}\), then PQ must also equal 8 for the triangle to be equilateral
- Strategy: Find which perimeter gives us \(\mathrm{PQ = 8}\)
3. SIMPLIFY the perimeter equation
- \(\mathrm{Perimeter = PR + RQ + PQ}\)
\(\mathrm{= 8 + 8 + PQ}\)
\(\mathrm{= 16 + PQ}\) - For equilateral triangle: \(\mathrm{PQ = 8}\)
- Therefore: \(\mathrm{Perimeter = 16 + 8 = 24}\) centimeters
4. APPLY CONSTRAINTS to verify our answer
- Check: If perimeter = 24, then \(\mathrm{PQ = 24 - 16 = 8}\) ✓
- This makes all sides equal \(\mathrm{(8, 8, 8)}\), confirming equilateral triangle
Answer: C (24 centimeters)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students fail to connect that "equilateral" means ALL sides must be equal, not just that we already have two equal sides.
They might think since we already have an isosceles triangle, any of the perimeter values could work, or they don't realize they need to work backwards to find what PQ would be. This leads to confusion and guessing among the answer choices.
Second Most Common Error:
Poor SIMPLIFY execution: Students set up the equation correctly but make arithmetic errors when solving for PQ from different perimeter values.
For example, they might miscalculate and think that a perimeter of 22 gives \(\mathrm{PQ = 8}\), when actually \(\mathrm{22 - 16 = 6}\). This computational error may lead them to select Choice A (22 centimeters).
The Bottom Line:
This problem tests whether students truly understand what makes a triangle equilateral versus just isosceles, and whether they can work backwards from perimeter to determine the third side length systematically.