Triangle JKL is similar to triangle MNP such that J corresponds to M, K corresponds to N, and L corresponds...

1. TRANSLATE the problem information

• Given information:
  • Triangle JKL is similar to triangle MNP
  • J corresponds to M, K corresponds to N, L corresponds to P
  • \(\mathrm{JK = MN}\) (these corresponding sides are equal)
  • \(\mathrm{JL = 9}\)
  • Need to find: MP

2. INFER the key relationship

• Since the triangles are similar, corresponding sides are proportional
• The crucial insight: JK corresponds to MN, and they're equal in length
• This means the ratio \(\mathrm{\frac{MN}{JK} = 1}\), so the scale factor is 1
• When the scale factor between similar triangles is 1, all corresponding sides are equal

3. INFER the final answer

• JL corresponds to MP
• Since scale factor = 1: \(\mathrm{MP = JL = 9}\)

Answer: C) 9

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students miss that equal corresponding sides means scale factor = 1

Many students recognize the triangles are similar and that sides are proportional, but they don't make the connection that when one pair of corresponding sides is equal, ALL corresponding sides must be equal. They might try to set up a proportion like \(\mathrm{\frac{JK}{MN} = \frac{JL}{MP}}\), substitute \(\mathrm{JK = MN}\), get \(\mathrm{1 = \frac{JL}{MP}}\), then solve incorrectly or get confused about what this means.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Conceptual confusion about similar triangles: Students think "similar" means "smaller version"

Some students assume that since one triangle is named after the other (JKL vs MNP), one must be smaller. They might think MP should be half of JL or some other fraction, leading them to select Choice A (3) or Choice B (4.5).

The Bottom Line:

This problem tests whether students understand that the scale factor between similar triangles can be 1 (making them congruent). The key insight is recognizing that equal corresponding sides immediately tells you the scale factor, eliminating the need for complex proportional reasoning.