In triangle ABC, the measure of angle A is 54°, the measure of angle B is 90°, and the measure...
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
In triangle ABC, the measure of angle A is \(54°\), the measure of angle B is \(90°\), and the measure of angle C is \(\frac{\mathrm{k}}{2}°\). What is the value of \(\mathrm{k}\)?
1. TRANSLATE the problem information
- Given information:
- Angle A = \(54°\)
- Angle B = \(90°\)
- Angle C = \(\frac{k}{2}°\)
- We need to find the value of k
2. INFER the approach
- Since we have a triangle with three angles, we can use the fundamental property that all triangle angles sum to \(180°\)
- Set up the equation: \(54° + 90° + \frac{k}{2}° = 180°\)
3. SIMPLIFY to solve for k
- First, combine the known angles: \(144° + \frac{k}{2}° = 180°\)
- Subtract \(144°\) from both sides: \(\frac{k}{2}° = 36°\)
- This means \(\frac{k}{2} = 36\)
- Multiply both sides by 2: \(k = 72\)
Answer: C. 72
Why Students Usually Falter on This Problem
Most Common Error Path:
Incomplete SIMPLIFY execution: Students solve correctly up to \(\frac{k}{2} = 36\), but then select this intermediate result as their final answer.
They get \(\frac{k}{2} = 36\) and think "Great, I have my answer!" without realizing they need to find k, not k/2. Since 36 appears as Choice A, this feels like confirmation they're correct.
This leads them to select Choice A (36) instead of continuing to multiply by 2.
The Bottom Line:
This problem tests whether students can complete a multi-step algebraic solution. The trap is that an intermediate step \(\left(\frac{k}{2} = 36\right)\) appears as a tempting answer choice, making it crucial to track what variable you're actually solving for.