Two similar right triangles have hypotenuses of length 15 and 45 respectively. The shorter leg of the smaller triangle measures...

1. TRANSLATE the problem information

• Given information:
  • Two similar right triangles
  • Smaller triangle: \(\mathrm{hypotenuse = 15}\), \(\mathrm{shorter\ leg = 9}\)
  • Larger triangle: \(\mathrm{hypotenuse = 45}\), shorter leg = unknown

• What this tells us: We need to find how much larger the second triangle is

2. INFER the solution approach

• Since the triangles are similar, all corresponding sides are in the same proportion
• We can find this proportion using any pair of corresponding sides - the hypotenuses work well since both are given

3. SIMPLIFY to find the scale factor

• \(\mathrm{Scale\ factor = \frac{larger\ hypotenuse}{smaller\ hypotenuse}}\)
• \(\mathrm{Scale\ factor = \frac{45}{15} = 3}\)
• This means the larger triangle is 3 times the size of the smaller triangle in every dimension

4. INFER and apply the scale factor to the shorter leg

• Since all sides scale by the same factor: \(\mathrm{shorter\ leg\ of\ larger\ triangle = 9 \times 3 = 27}\)

Answer: B (27)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that similar triangles have ALL sides proportional by the same ratio. They might try to use the Pythagorean theorem or get confused about which sides correspond to which, leading them to calculate incorrectly or get overwhelmed by unnecessary work.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misread "shorter leg" and might confuse which measurement belongs to which triangle, or they mix up the triangles' sizes. For instance, they might think the smaller triangle has the 45-unit hypotenuse.

This may lead them to select Choice A (18) by using an incorrect ratio or applying the ratio backwards.

The Bottom Line:

The key insight is recognizing that "similar triangles" is your cue to set up a proportion. Once you find the scale factor from any pair of corresponding sides, that same factor applies to ALL corresponding sides.