Question:Triangles PQR and STU are similar, with P, Q, R corresponding to S, T, U, respectively. The perimeter of triangle...
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
Question:
Triangles \(\mathrm{PQR}\) and \(\mathrm{STU}\) are similar, with \(\mathrm{P}\), \(\mathrm{Q}\), \(\mathrm{R}\) corresponding to \(\mathrm{S}\), \(\mathrm{T}\), \(\mathrm{U}\), respectively. The perimeter of triangle \(\mathrm{STU}\) is \(150\%\) of the perimeter of triangle \(\mathrm{PQR}\). If \(\mathrm{PQ} = 16\), what is the length of the corresponding side \(\mathrm{ST}\)?
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32
1. TRANSLATE the problem information
- Given information:
- Triangles PQR and STU are similar
- P corresponds to S, Q corresponds to T, R corresponds to U
- Perimeter of STU = 150% of perimeter of PQR
- PQ = 16
- Need to find ST
- What "150% of the perimeter" means: The perimeter of STU is 1.5 times the perimeter of PQR
2. INFER the key relationship
- In similar triangles, all corresponding sides are proportional with the same scale factor
- This same scale factor also applies to perimeters
- Since \(\mathrm{perimeter~of~STU} = 1.5 \times \mathrm{perimeter~of~PQR}\), the scale factor from PQR to STU is 1.5
3. SIMPLIFY to find the answer
- Since ST corresponds to PQ: \(\mathrm{ST} = 1.5 \times \mathrm{PQ}\)
- \(\mathrm{ST} = 1.5 \times 16 = 24\)
Answer: D (24)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students misinterpret "150% of the perimeter" as meaning the triangles have the same size, or they think it means STU's perimeter is 50% larger (adding 50% instead of recognizing 150% = 1.5 times).
This confusion about percentages leads them to use the wrong scale factor, possibly thinking the sides are equal or using 1.5 as an additive factor rather than multiplicative. This may lead them to select Choice A (16) if they think the sides are equal, or get confused and guess.
Second Most Common Error:
Missing conceptual knowledge: Students don't remember that in similar triangles, the ratio of corresponding sides equals the ratio of perimeters.
They might try to work backwards from the perimeter information but get stuck because they don't know how many sides the triangles have or what the other side lengths are. This causes them to get stuck and randomly select an answer.
The Bottom Line:
This problem tests whether students truly understand what "similar triangles" means beyond just "same shape" - specifically, that ALL measurements (sides, perimeters, etc.) scale by the same factor. The percentage conversion is also crucial but often trips students up.
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