Right triangles ABC and DEF are similar, where angles B and E are right angles and angle A corresponds to...

1. INFER the key relationship from similar triangles

• Given information:
  • Triangles ABC and DEF are similar
  • Angle A corresponds to angle D
  • \(\mathrm{cos(A) = \frac{12}{13}}\)

• Since the triangles are similar and angle A corresponds to angle D, these angles are equal: \(\mathrm{A = D}\)
• Therefore: \(\mathrm{tan(D) = tan(A)}\)

2. INFER what the cosine value tells us about the triangle

• \(\mathrm{cos(A) = \frac{12}{13}}\) means \(\mathrm{\frac{adjacent}{hypotenuse} = \frac{12}{13}}\)
• We can think of this as a right triangle where:
  • Adjacent side (to angle A) = 12
  • Hypotenuse = 13
  • Opposite side = unknown

3. SIMPLIFY using the Pythagorean theorem to find the missing side

• Apply \(\mathrm{a^2 + b^2 = c^2}\):
  \(\mathrm{opposite^2 + 12^2 = 13^2}\)
  \(\mathrm{opposite^2 + 144 = 169}\)
  \(\mathrm{opposite^2 = 25}\)
  \(\mathrm{opposite = 5}\)

4. SIMPLIFY to calculate the tangent ratio

• \(\mathrm{tan(A) = \frac{opposite}{adjacent} = \frac{5}{12}}\)
• Since \(\mathrm{tan(D) = tan(A)}\), we have \(\mathrm{tan(D) = \frac{5}{12}}\)

Answer: B (5/12)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that corresponding angles in similar triangles are equal, so they get confused about how to connect angle A information to angle D.

Without this connection, they might try to work backwards from the answer choices or attempt to use the cosine value directly as the tangent value. This leads to confusion and guessing.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly identify that they need to find tan(A) but make arithmetic errors when applying the Pythagorean theorem (like getting \(\mathrm{opposite = \sqrt{25} = 5}\) wrong) or when setting up the tangent ratio.

This may lead them to select Choice A (5/13) if they incorrectly calculate tan as opposite/hypotenuse instead of opposite/adjacent.

The Bottom Line:

This problem tests whether students can bridge the gap between similar triangles (a geometry concept) and trigonometry. The key insight is recognizing that "corresponding angles are equal" is the bridge that allows us to transfer trigonometric information from one triangle to another.