Triangle ABC is similar to triangle DEF, where angle A corresponds to angle D and angles C and F are...

1. TRANSLATE the problem information

• Given information:
  • Triangle ABC ~ Triangle DEF (similar triangles)
  • Angle A corresponds to angle D
  • Angles C and F are right angles (90°)
  • \(\mathrm{AB = 2.9 \times DE}\) (this information isn't needed for our solution)
  • \(\mathrm{tan\,A = \frac{21}{20}}\)
  • Find: sin D

2. INFER the key relationship

• Since the triangles are similar, corresponding angles are equal
• This means: \(\mathrm{tan\,A = tan\,D = \frac{21}{20}}\)
• Strategy: Use this tangent value to find the sides of triangle DEF, then calculate sin D

3. INFER the side relationships

• In right triangle DEF, \(\mathrm{tan\,D = \frac{opposite}{adjacent} = \frac{EF}{DF} = \frac{21}{20}}\)
• We can think of the sides as \(\mathrm{EF = 21k}\) and \(\mathrm{DF = 20k}\) for some positive value k
• The actual value of k doesn't matter since we're looking for ratios

4. SIMPLIFY to find the hypotenuse

• Use Pythagorean theorem: \(\mathrm{DE^2 = EF^2 + DF^2}\)
• \(\mathrm{DE^2 = (21k)^2 + (20k)^2}\)
• \(\mathrm{DE^2 = 441k^2 + 400k^2 = 841k^2}\)
• \(\mathrm{DE = \sqrt{841k^2} = 29k}\)

5. SIMPLIFY to find sin D

• \(\mathrm{sin\,D = \frac{opposite}{hypotenuse} = \frac{EF}{DE}}\)
• \(\mathrm{sin\,D = \frac{21k}{29k} = \frac{21}{29}}\)
• Converting to decimal (use calculator): \(\mathrm{21 \div 29 = 0.7241}\)

Answer: 21/29, 0.7241, or 0.724

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students fail to recognize that similar triangles have equal corresponding angles, so they don't establish that tan A = tan D. Instead, they might try to use the given scale factor (2.9) or attempt to work directly with triangle ABC without connecting it to triangle DEF. This leads to confusion about how to proceed and results in guessing.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly identify that tan D = 21/20 but make errors when applying the Pythagorean theorem. They might calculate DE² = 21² + 20² = 441 + 400 = 841 correctly, but then make arithmetic errors finding \(\mathrm{\sqrt{841} = 29}\). This leads to an incorrect hypotenuse value and therefore an incorrect sine ratio.

The Bottom Line:

This problem tests whether students can bridge the gap between similar triangles and trigonometry. The key insight is recognizing that similarity preserves angle relationships, allowing us to transfer the tangent value from one triangle to its corresponding triangle. Once that connection is made, it becomes a straightforward trigonometry problem.