In triangle ABC, side AB has length 6, side AC has length 9, and the measure of angle A is...
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
In triangle \(\mathrm{ABC}\), side \(\mathrm{AB}\) has length \(\mathrm{6}\), side \(\mathrm{AC}\) has length \(\mathrm{9}\), and the measure of angle \(\mathrm{A}\) is \(\mathrm{52°}\). In triangle \(\mathrm{DEF}\), side \(\mathrm{DE}\) has length \(\mathrm{4}\), side \(\mathrm{DF}\) has length \(\mathrm{6}\), and the measure of angle \(\mathrm{D}\) is \(\mathrm{52°}\). Which of the following additional pieces of information is needed to determine whether triangle \(\mathrm{ABC}\) is similar to triangle \(\mathrm{DEF}\)?
The measure of angle B
The measure of angle E
The lengths of sides BC and EF
No additional information is needed
1. TRANSLATE the problem information
- Given information:
- Triangle ABC: \(\mathrm{AB = 6}\), \(\mathrm{AC = 9}\), \(\mathrm{angle\ A = 52°}\)
- Triangle DEF: \(\mathrm{DE = 4}\), \(\mathrm{DF = 6}\), \(\mathrm{angle\ D = 52°}\)
- Need to determine: What additional information is needed for similarity?
2. INFER the appropriate similarity criterion
- Since we have two sides and an included angle for each triangle, we can use SAS similarity
- For SAS similarity, we need: two pairs of proportional sides with equal included angles
- The angles A and D are both \(\mathrm{52°}\), so they're equal ✓
3. INFER which sides to compare
- The included angle is formed by two sides
- In triangle ABC: angle A is formed by sides AB and AC
- In triangle DEF: angle D is formed by sides DE and DF
- These are the corresponding side pairs we need to check
4. SIMPLIFY by computing the ratios
- First ratio: \(\mathrm{AB/DE = 6/4 = 3/2}\)
- Second ratio: \(\mathrm{AC/DF = 9/6 = 3/2}\)
- Both ratios are equal!
5. INFER the conclusion
- Since the ratios are equal (\(\mathrm{3/2}\)) and the included angles are equal (\(\mathrm{52°}\)), SAS similarity is satisfied
- No additional information is needed
Answer: D
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Not recognizing that SAS similarity criterion applies here
Students often think they need all three angles or all three sides to prove similarity. They might look at the answer choices and think "I need angle B" or "I need the third sides" without realizing that two proportional sides with an included angle is sufficient.
This may lead them to select Choice A (The measure of angle B) or Choice C (The lengths of sides BC and EF)
Second Most Common Error:
Conceptual confusion about included angles: Not understanding which angle is "included"
Some students might think any angle can be used with any two sides, not realizing that the angle must be between the two given sides. They might incorrectly try to use angle A with sides AB and some other side.
This leads to confusion about whether they have enough information, causing them to guess among the choices.
The Bottom Line:
This problem tests whether students know similarity criteria beyond just "all angles equal" or "all sides proportional." The key insight is recognizing that SAS similarity (two proportional sides with equal included angle) is sufficient to establish similarity.
The measure of angle B
The measure of angle E
The lengths of sides BC and EF
No additional information is needed