In right triangle ABC with right angle at C, the altitude from C to hypotenuse AB intersects AB at point...

1. TRANSLATE the problem information

• Given information:
  • Right triangle ABC with right angle at C
  • CD is altitude from C to hypotenuse AB
  • \(\mathrm{AD:DB = 9:16}\)
  • \(\mathrm{CD = 12}\)
  • Need to find sin A

2. INFER the approach using ratios

• Since \(\mathrm{AD:DB = 9:16}\), we can write \(\mathrm{AD = 9k}\) and \(\mathrm{DB = 16k}\) for some positive value k
• This means \(\mathrm{AB = AD + DB = 9k + 16k = 25k}\)
• The key insight is that we can use the altitude-to-hypotenuse theorem to find k

3. INFER and apply the altitude-to-hypotenuse theorem

• The theorem states: \(\mathrm{CD^2 = AD \times DB}\)
• Substituting our values: \(\mathrm{12^2 = (9k)(16k)}\)
• This gives us: \(\mathrm{144 = 144k^2}\)

4. SIMPLIFY to find k

• From \(\mathrm{144 = 144k^2}\)
• Divide both sides by 144: \(\mathrm{k^2 = 1}\)
• Therefore \(\mathrm{k = 1}\) (taking positive value since lengths are positive)
• This means \(\mathrm{AD = 9}\), \(\mathrm{DB = 16}\), and \(\mathrm{AB = 25}\)

5. INFER how to find the triangle legs

• Use the altitude theorems:
  • \(\mathrm{AC^2 = AD \times AB}\)
  • \(\mathrm{BC^2 = DB \times AB}\)
• These relationships come from the similar triangles created by the altitude

6. SIMPLIFY to find the legs

• \(\mathrm{AC^2 = AD \times AB = 9 \times 25 = 225}\), so \(\mathrm{AC = 15}\)
• \(\mathrm{BC^2 = DB \times AB = 16 \times 25 = 400}\), so \(\mathrm{BC = 20}\)

7. APPLY CONSTRAINTS and verify

• Check using Pythagorean theorem: \(\mathrm{15^2 + 20^2 = 225 + 400 = 625 = 25^2}\) ✓
• This confirms our triangle is valid

8. INFER which sides to use for sin A

• \(\mathrm{sin\,A = \frac{opposite\,side}{hypotenuse} = \frac{BC}{AB} = \frac{20}{25} = \frac{4}{5}}\)

Answer: \(\mathrm{4/5}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that they should apply the altitude-to-hypotenuse theorem to find k. Instead, they might try to use the Pythagorean theorem directly with the given ratio, leading to equations with multiple unknowns that can't be solved. This causes them to get stuck and abandon systematic solution, leading to guessing.

Second Most Common Error:

Missing conceptual knowledge: Students may not remember the altitude theorems (\(\mathrm{AC^2 = AD \times AB}\) and \(\mathrm{BC^2 = DB \times AB}\)) needed to find the triangle legs. They might try to use other relationships or geometric properties that don't apply, leading to incorrect leg lengths and ultimately the wrong sine ratio. This may lead them to select an incorrect fractional answer.

The Bottom Line:

This problem requires deep understanding of the special relationships created when an altitude is drawn to the hypotenuse of a right triangle. Success depends on recognizing which geometric theorems apply and applying them in the correct sequence.