In the xy-plane, a right triangle ABC has vertices at \(\mathrm{A(2,1)}\), \(\mathrm{B(14,1)}\), and \(\mathrm{C(14,10)}\). A second triangle, DEF, is...
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
In the xy-plane, a right triangle \(\mathrm{ABC}\) has vertices at \(\mathrm{A(2,1)}\), \(\mathrm{B(14,1)}\), and \(\mathrm{C(14,10)}\). A second triangle, \(\mathrm{DEF}\), is similar to triangle \(\mathrm{ABC}\) with the same orientation, where vertices \(\mathrm{D}\), \(\mathrm{E}\), and \(\mathrm{F}\) correspond to vertices \(\mathrm{A}\), \(\mathrm{B}\), and \(\mathrm{C}\), respectively. What is the slope of the segment \(\mathrm{DF}\)?
\(-\frac{4}{3}\)
\(\frac{3}{4}\)
\(\frac{5}{4}\)
\(\frac{4}{3}\)
1. TRANSLATE the correspondence information
- Given information:
- Triangle ABC has vertices \(\mathrm{A(2,1), B(14,1), C(14,10)}\)
- Triangle DEF is similar to ABC with same orientation
- D corresponds to A, E corresponds to B, F corresponds to C
- Need to find slope of segment DF
2. INFER which segments correspond
- Since D corresponds to A and F corresponds to C, segment DF corresponds to segment AC
- This is the key insight: we need to find the slope of AC to determine the slope of DF
3. INFER the relationship between corresponding segments
- In similar triangles with the same orientation, corresponding segments are parallel
- Since DF corresponds to AC, these segments are parallel
- Parallel lines have equal slopes, so slope of DF = slope of AC
4. SIMPLIFY by calculating the slope of AC
- Using points \(\mathrm{A(2,1)}\) and \(\mathrm{C(14,10)}\) with the slope formula:
- Slope of AC = \(\mathrm{\frac{10-1}{14-2}}\)
- \(\mathrm{= \frac{9}{12}}\)
- \(\mathrm{= \frac{3}{4}}\)
Answer: B. 3/4
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students don't recognize that DF corresponds to AC based on the given correspondences. Instead, they might try to calculate slopes of other segments like AB or BC, or they might not understand what "corresponding segments" means in the context of similar triangles.
This leads to confusion and guessing among the answer choices.
Second Most Common Error:
Poor TRANSLATE reasoning: Students misinterpret which vertices correspond to which. They might think they need to find slopes of segments that don't actually correspond to DF, such as assuming DF corresponds to AB or BC instead of AC.
This may lead them to calculate wrong slopes and select incorrect answer choices.
The Bottom Line:
This problem tests whether students understand the correspondence relationship in similar triangles and can apply the property that corresponding segments in similar triangles with the same orientation are parallel. The geometric insight about which segments correspond is more crucial than the slope calculation itself.
\(-\frac{4}{3}\)
\(\frac{3}{4}\)
\(\frac{5}{4}\)
\(\frac{4}{3}\)