Question:Triangles ABC and DEF are similar, where vertices A, B, and C correspond to vertices D, E, and F, respectively....
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
Triangles \(\mathrm{ABC}\) and \(\mathrm{DEF}\) are similar, where vertices \(\mathrm{A}\), \(\mathrm{B}\), and \(\mathrm{C}\) correspond to vertices \(\mathrm{D}\), \(\mathrm{E}\), and \(\mathrm{F}\), respectively. In triangle \(\mathrm{ABC}\), the length of side \(\mathrm{AB}\) is 6 and the length of side \(\mathrm{BC}\) is 8. The perimeter of triangle \(\mathrm{ABC}\) is 24. If the length of side \(\mathrm{DE}\) is 9, what is the length of side \(\mathrm{EF}\)?
Answer Choices:
- 8
- 10
- 12
- 15
1. TRANSLATE the correspondence information
- Given information:
- Triangle ABC ~ Triangle DEF
- A corresponds to D, B corresponds to E, C corresponds to F
- \(\mathrm{AB = 6}\), \(\mathrm{BC = 8}\), \(\mathrm{perimeter\,of\,ABC = 24}\), \(\mathrm{DE = 9}\)
- This tells us that AB corresponds to DE, and BC corresponds to EF
2. TRANSLATE the perimeter to find the missing side
- \(\mathrm{Perimeter = AB + BC + AC = 24}\)
- \(\mathrm{6 + 8 + AC = 24}\)
- \(\mathrm{AC = 10}\)
3. INFER the approach using similar triangles
- Since the triangles are similar, corresponding sides are proportional
- We can use AB and DE to find the scale factor
- Then apply this same scale factor to BC to find EF
4. SIMPLIFY to find the ratio
- Scale factor = \(\mathrm{\frac{DE}{AB} = \frac{9}{6} = \frac{3}{2}}\)
- This means triangle DEF is 1.5 times larger than triangle ABC
5. SIMPLIFY to find EF
- \(\mathrm{EF = BC \times scale\,factor}\)
- \(\mathrm{EF = 8 \times \frac{3}{2} = 12}\)
Answer: C. 12
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skills: Students misinterpret the correspondence, thinking BC corresponds to DE instead of EF.
They might set up the ratio as \(\mathrm{\frac{BC}{DE} = \frac{8}{9}}\), then try to find EF using a different relationship. This leads to confusion about which sides actually correspond to each other, causing them to get stuck and guess randomly.
Second Most Common Error:
Poor SIMPLIFY execution: Students correctly identify that \(\mathrm{EF = BC \times \frac{3}{2}}\) but make arithmetic errors.
Some might calculate \(\mathrm{8 \times \frac{3}{2}}\) as \(\mathrm{8 \times 3 = 24}\), then forget to divide by 2, or make other fraction multiplication errors. This could lead them to select Choice D (15) if they incorrectly compute the ratio or apply it wrong.
The Bottom Line:
The key insight is recognizing that similar triangles create a consistent scale factor between ALL corresponding pairs - once you find the ratio using one pair, it applies to every other pair. Students who miss this proportional relationship struggle to connect the given information systematically.