In triangleABC, angleB is a right angle and the length of BC is 136 millimeters. If cosA = 3/5, what...

1. TRANSLATE the problem information

• Given information:
  • Triangle ABC with right angle at B
  • BC = 136 mm
  • \(\cos \mathrm{A} = \frac{3}{5}\)
  • Find: length of AB

2. INFER the trigonometric relationship

• Since \(\cos \mathrm{A} = \frac{3}{5}\), and \(\cos = \frac{\mathrm{adjacent}}{\mathrm{hypotenuse}}\)
• AB is adjacent to angle A, AC is the hypotenuse
• This means \(\frac{\mathrm{AB}}{\mathrm{AC}} = \frac{3}{5}\)
• We can represent \(\mathrm{AB} = 3\mathrm{k}\) and \(\mathrm{AC} = 5\mathrm{k}\) where k is a scale factor

3. INFER the strategy using Pythagorean theorem

• In right triangle ABC: \(\mathrm{AB}^2 + \mathrm{BC}^2 = \mathrm{AC}^2\)
• Substitute our expressions: \((3\mathrm{k})^2 + 136^2 = (5\mathrm{k})^2\)

4. SIMPLIFY the equation to find k

• \((3\mathrm{k})^2 + 136^2 = (5\mathrm{k})^2\)
• \(9\mathrm{k}^2 + 18{,}496 = 25\mathrm{k}^2\)
• \(18{,}496 = 16\mathrm{k}^2\)
• \(\mathrm{k}^2 = 1{,}156\)
• \(\mathrm{k} = 34\) (use calculator if needed)

5. SIMPLIFY to find AB

• \(\mathrm{AB} = 3\mathrm{k} = 3(34) = 102\) mm

Answer: B. 102

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse which side is adjacent to angle A versus which is opposite. They might think BC is adjacent to angle A instead of AB.

If they incorrectly use BC as the adjacent side, they would set up \(\cos \mathrm{A} = \frac{\mathrm{BC}}{\mathrm{AC}} = \frac{136}{\mathrm{AC}} = \frac{3}{5}\), leading to \(\mathrm{AC} = 136 \times \frac{5}{3} \approx 227\). Then using Pythagorean theorem incorrectly would lead to completely wrong values. This leads to confusion and guessing.

Second Most Common Error:

Poor INFER reasoning: Students recognize the correct trigonometric setup but fail to use the parametric approach (3k and 5k). Instead, they try to solve directly for individual sides without establishing the proportional relationship.

Without the scale factor insight, they get stuck trying to work with two unknowns (AB and AC) and only one equation from the cosine relationship. This may lead them to select Choice A (34) if they incorrectly identify 34 as a side length rather than the scale factor.

The Bottom Line:

This problem requires connecting trigonometric ratios with the Pythagorean theorem through proportional reasoning - a sophisticated multi-step process that challenges students to see relationships between different mathematical concepts.