Triangles A'B'C' and ABC are similar, with A corresponding to A', B corresponding to B', and C corresponding to C'....
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
Triangles \(\mathrm{A'B'C'}\) and \(\mathrm{ABC}\) are similar, with \(\mathrm{A}\) corresponding to \(\mathrm{A'}\), \(\mathrm{B}\) corresponding to \(\mathrm{B'}\), and \(\mathrm{C}\) corresponding to \(\mathrm{C'}\). In triangle \(\mathrm{ABC}\), the measure of angle \(\mathrm{A}\) is \(37°\), and the measure of angle \(\mathrm{B}\) is \(53°\). The area of triangle \(\mathrm{A'B'C'}\) is \(9\) times the area of triangle \(\mathrm{ABC}\). What is the measure of angle \(\mathrm{C'}\)?
\(30°\)
\(37°\)
\(53°\)
\(90°\)
1. TRANSLATE the problem information
- Given information:
- Triangles ABC and A'B'C' are similar
- Angle A = 37°, Angle B = 53° (in triangle ABC)
- Area of A'B'C' is 9 times area of ABC
- Need to find angle C'
2. INFER the solution approach
- Key insight: In similar triangles, corresponding angles are equal
- To find angle C', we first need to find angle C in triangle ABC
- The area ratio information won't affect angle measures
3. Find angle C using angle sum property
- Angles in triangle ABC must sum to 180°:
\(\mathrm{Angle\ C = 180° - (Angle\ A + Angle\ B)}\)
\(\mathrm{Angle\ C = 180° - (37° + 53°)}\)
\(\mathrm{Angle\ C = 180° - 90° = 90°}\)
4. INFER the final answer using similarity
- Since the triangles are similar and C corresponds to C':
\(\mathrm{Angle\ C' = Angle\ C = 90°}\)
Answer: D) \(\mathrm{90°}\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Getting distracted by the area ratio information and thinking it affects angle measures.
Students might try to use the fact that the area is "9 times larger" to somehow calculate angles, not realizing that similarity affects side lengths and areas, but corresponding angles remain equal regardless of scale factor. This leads to confusion about which formulas to use and may cause them to abandon systematic solution and guess.
Second Most Common Error:
Missing conceptual knowledge: Not remembering that angles in a triangle sum to 180°.
Without this basic property, students cannot find angle C and get stuck early in the problem. They might recognize that similar triangles have equal corresponding angles, but can't proceed to find what angle C equals. This causes them to get stuck and randomly select an answer.
The Bottom Line:
This problem tests whether students can identify relevant vs. irrelevant information and apply basic triangle properties systematically. The area scaling information is a red herring designed to test focus and understanding of what similarity actually preserves.
\(30°\)
\(37°\)
\(53°\)
\(90°\)