Two similar right triangles, Triangle FGH and Triangle JKL, have areas of 12 square units and 108 square units, respectively....

1. TRANSLATE the problem information

• Given information:
  • Triangle FGH and Triangle JKL are similar right triangles
  • Area of Triangle FGH = 12 square units
  • Area of Triangle JKL = 108 square units
  • Hypotenuse of Triangle FGH = 5 units
  • Need to find: Hypotenuse of Triangle JKL

2. INFER the key relationship

• Since the triangles are similar, their corresponding sides are proportional by some scaling factor k
• Critical insight: For similar figures, areas don't scale the same way as sides
• If sides scale by factor k, then areas scale by factor \(\mathrm{k^2}\)
• This means: \(\mathrm{\frac{Area_{JKL}}{Area_{FGH}} = k^2}\)

3. SIMPLIFY to find the scaling factor

• Set up the equation: \(\mathrm{k^2 = \frac{Area_{JKL}}{Area_{FGH}} = \frac{108}{12} = 9}\)
• Take the square root to find k: \(\mathrm{k = \sqrt{9} = 3}\)
• This tells us each side of Triangle JKL is 3 times the corresponding side of Triangle FGH

4. Apply the scaling factor

• Hypotenuse of Triangle JKL = \(\mathrm{k \times Hypotenuse_{FGH}}\)
• Hypotenuse of Triangle JKL = \(\mathrm{3 \times 5 = 15}\) units

Answer: A (15)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students miss the crucial distinction between how sides and areas scale in similar figures. They incorrectly assume that if one triangle has 9 times the area (\(\mathrm{108 \div 12 = 9}\)), then its sides are also 9 times longer. Using \(\mathrm{k = 9}\), they calculate the hypotenuse as \(\mathrm{9 \times 5 = 45}\).

This may lead them to select Choice C (45).

Second Most Common Error:

Conceptual confusion about similarity: Students may not realize the triangles are similar or may not know how to use this information systematically. They might try to work backwards from the answer choices or attempt to use the individual areas without understanding the scaling relationship.

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

The key challenge is recognizing that similarity creates a squared relationship between area ratios and side ratios. Students must take the square root of the area ratio to find how the sides actually scale.