Triangles ABC and DEF are similar, with A, B, and C corresponding to D, E, and F, respectively. The perimeters...
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
Triangles \(\mathrm{ABC}\) and \(\mathrm{DEF}\) are similar, with \(\mathrm{A}\), \(\mathrm{B}\), and \(\mathrm{C}\) corresponding to \(\mathrm{D}\), \(\mathrm{E}\), and \(\mathrm{F}\), respectively. The perimeters of triangle \(\mathrm{ABC}\) and triangle \(\mathrm{DEF}\) are equal. The measure of angle \(\mathrm{F}\) is \(35°\). What is the measure of angle \(\mathrm{C}\)?
1. TRANSLATE the problem information
- Given information:
- Triangles ABC and DEF are similar
- A corresponds to D, B corresponds to E, C corresponds to F
- Perimeters are equal
- Angle F = 35°
- Find angle C
2. INFER the key relationship
- The most important property of similar triangles: corresponding angles are always equal
- This is true regardless of how big or small the triangles are
- Since the problem tells us C corresponds to F, we know \(\angle \mathrm{C} = \angle \mathrm{F}\)
3. Apply the correspondence
- We know \(\angle \mathrm{F} = 35°\)
- Since \(\angle \mathrm{C} = \angle \mathrm{F}\) (corresponding angles)
- Therefore \(\angle \mathrm{C} = 35°\)
Answer: C (35°)
Why Students Usually Falter on This Problem
Most Common Error Path:
Conceptual confusion about similar triangles: Students might think that since the perimeters are equal, something special must be calculated, or they might confuse similar triangles with congruent triangles and think additional work is needed.
Some students don't realize that corresponding angles in similar triangles are always equal, thinking instead that only the ratios of sides are preserved. This leads to confusion and guessing among the answer choices.
Second Most Common Error:
Weak TRANSLATE skill: Students may not properly understand what "A, B, and C corresponding to D, E, and F, respectively" means, and therefore can't identify which angles are corresponding.
Without knowing that C corresponds to F, they can't make the direct connection that \(\angle \mathrm{C} = \angle \mathrm{F}\). This may lead them to select Choice A (15°) or Choice B (25°) by trying to guess a relationship with the given 35°.
The Bottom Line:
This problem tests whether students truly understand what "similar triangles" means. The equal perimeters are a red herring - the key insight is simply that corresponding angles are equal, period.