Question:In triangle PQR, side QR has length 180. Point S lies on side PR, and a line through S parallel...
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
In triangle PQR, side QR has length 180. Point S lies on side PR, and a line through S parallel to QR intersects side PQ at point T, forming triangle PST inside triangle PQR. If \(\mathrm{QT = 4 \cdot PT}\), what is the length of ST?
1. TRANSLATE the problem information
- Given information:
- Triangle PQR with \(\mathrm{QR = 180}\)
- Point S lies on side PR
- Line through S parallel to QR intersects PQ at point T
- \(\mathrm{QT = 4 \cdot PT}\)
- Need to find: length of \(\mathrm{ST}\)
2. INFER the geometric relationship
- Since ST is parallel to QR, triangles PST and PQR share vertex P and have parallel corresponding sides
- This creates similar triangles: \(\triangle\mathrm{PST} \sim \triangle\mathrm{PQR}\)
- Similar triangles have proportional corresponding sides
3. TRANSLATE the given condition into a useful ratio
- Given: \(\mathrm{QT = 4 \cdot PT}\)
- Since \(\mathrm{PQ = PT + QT}\), we have:
\(\mathrm{PQ = PT + 4PT = 5PT}\) - Therefore: \(\mathrm{PT/PQ = PT/(5PT) = 1/5}\)
4. INFER how to apply similarity
- Since \(\triangle\mathrm{PST} \sim \triangle\mathrm{PQR}\), corresponding sides are proportional:
\(\mathrm{ST/QR = PS/PR = PT/PQ}\) - We know \(\mathrm{PT/PQ = 1/5}\), so \(\mathrm{ST/QR = 1/5}\)
5. SIMPLIFY to find ST
- \(\mathrm{ST/QR = 1/5}\)
- \(\mathrm{ST/180 = 1/5}\)
- \(\mathrm{ST = (1/5) \times 180 = 36}\)
Answer: 36
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students fail to recognize that the parallel line creates similar triangles, instead trying to use other geometric relationships like the Pythagorean theorem or attempting to calculate areas. Without recognizing similarity, they get stuck trying various unrelated approaches and end up guessing.
Second Most Common Error:
Poor TRANSLATE reasoning: Students correctly identify similar triangles but misinterpret the relationship \(\mathrm{QT = 4 \cdot PT}\). They might think \(\mathrm{PT/QT = 1/4}\) and incorrectly conclude that \(\mathrm{PT/PQ = 1/4}\) instead of properly recognizing that \(\mathrm{PQ = PT + QT = 5PT}\), leading to \(\mathrm{PT/PQ = 1/5}\). This error would give them \(\mathrm{ST = (1/4) \times 180 = 45}\).
The Bottom Line:
This problem tests whether students can recognize the powerful connection between parallel lines and similar triangles, then correctly translate a given ratio into the similarity scale factor needed for the solution.