In right triangle ABC, angle C is the right angle and BC = 162. Point D on side AB is...
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
In right triangle \(\mathrm{ABC}\), angle \(\mathrm{C}\) is the right angle and \(\mathrm{BC = 162}\). Point \(\mathrm{D}\) on side \(\mathrm{AB}\) is connected by a line segment with point \(\mathrm{E}\) on side \(\mathrm{AC}\) such that line segment \(\mathrm{DE}\) is parallel to side \(\mathrm{BC}\) and \(\mathrm{CE = 2AE}\). What is the length of line segment \(\mathrm{DE}\)?
1. TRANSLATE the problem information
- Given information:
- Right triangle ABC with right angle at C
- \(\mathrm{BC = 162}\)
- Point D is on side AB, point E is on side AC
- DE is parallel to BC
- \(\mathrm{CE = 2AE}\)
2. INFER the key geometric relationship
- Since DE is parallel to BC, we have parallel lines cut by transversals AB and AC
- This creates corresponding angles: angle ADE = angle ABC and angle AED = angle ACB
- By the AA similarity postulate, triangle ADE is similar to triangle ABC
3. TRANSLATE the segment relationship into a ratio
- Since E lies on AC: \(\mathrm{AE + CE = AC}\)
- Given that \(\mathrm{CE = 2AE}\): \(\mathrm{AE + 2AE = AC}\)
- SIMPLIFY: \(\mathrm{3AE = AC}\), so \(\mathrm{AE/AC = 1/3}\)
4. INFER and apply the similarity property
- Since triangle ADE ~ triangle ABC, corresponding sides are proportional
- This means: \(\mathrm{DE/BC = AE/AC = AD/AB}\)
5. SIMPLIFY to find DE
- We know \(\mathrm{DE/BC = AE/AC = 1/3}\)
- Substituting \(\mathrm{BC = 162}\): \(\mathrm{DE/162 = 1/3}\)
- Therefore: \(\mathrm{DE = 162 \times (1/3) = 54}\)
Answer: 54
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students fail to recognize that DE || BC creates similar triangles. They might try to use the Pythagorean theorem directly or attempt to find individual side lengths without establishing the similarity relationship. Without this key insight, they cannot access the proportional relationships needed to solve the problem, leading to confusion and guessing.
Second Most Common Error:
Poor TRANSLATE reasoning: Students correctly identify the similar triangles but misinterpret the condition \(\mathrm{CE = 2AE}\). They might incorrectly conclude that \(\mathrm{AE/AC = 2/3}\) instead of 1/3, thinking that since CE is twice AE, then AE must be the larger portion. This leads them to calculate \(\mathrm{DE = 162 \times (2/3) = 108}\), which would be an incorrect answer.
The Bottom Line:
This problem requires students to connect the geometric concept of parallel lines with similarity, then carefully translate the given segment relationship into the correct ratio. The key breakthrough is recognizing that parallel segments create similar triangles, making this fundamentally a similarity problem rather than a direct measurement problem.