Triangle ABC is similar to triangle DEF. The area of triangle ABC is 324 square inches, and the area of...

1. TRANSLATE the problem information

• Given information:
  • Triangle ABC is similar to triangle DEF
  • Area of ABC = 324 square inches
  • Area of DEF = 36 square inches
  • Height of ABC = 27 inches
  • Need to find: height of DEF

2. INFER the key relationship for similar figures

• Since the triangles are similar, their corresponding sides are proportional
• For similar figures, the ratio of their areas equals the square of the ratio of their corresponding linear dimensions
• This means: \(\mathrm{Area\,ratio = (Linear\,ratio)^2}\)

3. SIMPLIFY to find the area ratio

• Area ratio = Area of ABC ÷ Area of DEF
• \(\mathrm{Area\,ratio = 324 \div 36 = 9}\)

4. INFER and SIMPLIFY to find the linear ratio

• Since \(\mathrm{Area\,ratio = (Linear\,ratio)^2}\)
• We have: \(\mathrm{(Linear\,ratio)^2 = 9}\)
• Taking the square root: \(\mathrm{Linear\,ratio = \sqrt{9} = 3}\)
• This means each dimension of ABC is 3 times the corresponding dimension of DEF

5. SIMPLIFY to find the height of DEF

• Height of ABC = 3 × Height of DEF
• \(\mathrm{27 = 3 \times Height\,of\,DEF}\)
• \(\mathrm{Height\,of\,DEF = 27 \div 3 = 9\,inches}\)

Answer: C (9)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the area-to-linear ratio relationship for similar figures. Instead, they might think the area ratio directly equals the linear ratio, reasoning that if areas are in a 9:1 ratio, then corresponding sides are also in a 9:1 ratio. Using this incorrect reasoning: \(\mathrm{Height\,of\,DEF = 27 \div 9 = 3\,inches}\).

This may lead them to select Choice A (3).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify that they need the square root of the area ratio but make an arithmetic error. They might calculate the area ratio as \(\mathrm{324 \div 36 = 8}\) instead of 9, then take \(\mathrm{\sqrt{8} \approx 2.8}\), leading to \(\mathrm{Height\,of\,DEF = 27 \div 2.8 \approx 9.6}\), which they round to the nearest answer choice.

This may lead them to select Choice D (12) or cause confusion and guessing.

The Bottom Line:

The key insight is recognizing that area scales by the square of the linear scale factor for similar figures. Students who miss this fundamental relationship will struggle with any similar triangles problem involving areas.