In triangle ABC, angle B is a right angle. The length of side AB is 10sqrt(37) and the length of...

1. TRANSLATE the problem information

• Given information:
  • Triangle ABC has a right angle at B
  • Side \(\mathrm{AB = 10\sqrt{37}}\)
  • Side \(\mathrm{BC = 24\sqrt{37}}\)
  • Need to find side AC

2. INFER the geometric relationship

• Since angle B is the right angle, side AC is the hypotenuse
• Sides AB and BC are the legs of the right triangle
• This means we can use the Pythagorean theorem: \(\mathrm{c^2 = a^2 + b^2}\)

3. TRANSLATE into the Pythagorean equation

• Set up: \(\mathrm{AC^2 = (AB)^2 + (BC)^2}\)
• Substitute: \(\mathrm{AC^2 = (10\sqrt{37})^2 + (24\sqrt{37})^2}\)

4. SIMPLIFY the calculation

• Calculate each squared term:
  • \(\mathrm{(10\sqrt{37})^2 = 10^2 \times (\sqrt{37})^2 = 100 \times 37 = 3,700}\)
  • \(\mathrm{(24\sqrt{37})^2 = 24^2 \times (\sqrt{37})^2 = 576 \times 37 = 21,312}\)
• Add: \(\mathrm{AC^2 = 3,700 + 21,312 = 25,012}\)
• Factor: \(\mathrm{AC^2 = 37 \times (100 + 576) = 37 \times 676}\)

5. SIMPLIFY to find the final answer

• Take the square root: \(\mathrm{AC = \sqrt{37 \times 676} = \sqrt{37} \times \sqrt{676} = \sqrt{37} \times 26 = 26\sqrt{37}}\)

Answer: B. \(\mathrm{26\sqrt{37}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not correctly identifying which side is the hypotenuse in the right triangle

Students may assume that one of the given sides (AB or BC) is the hypotenuse simply because they're given, rather than recognizing that the hypotenuse is always opposite the right angle. Since angle B is the right angle, AC must be the hypotenuse. This error leads to incorrectly setting up equations like \(\mathrm{AB^2 = AC^2 + BC^2}\) or \(\mathrm{BC^2 = AB^2 + AC^2}\), producing completely wrong numerical results that don't match any answer choice.

This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Making arithmetic errors when computing \(\mathrm{(10\sqrt{37})^2 + (24\sqrt{37})^2}\)

Students might incorrectly calculate \(\mathrm{10^2 = 1000}\) instead of 100, or \(\mathrm{24^2 = 576}\) but then make addition errors, or forget that \(\mathrm{(a\sqrt{b})^2 = a^2b}\). These computational errors lead to wrong values under the radical, preventing them from reaching the correct answer.

This may lead them to select wrong answer choices or abandon the systematic solution.

The Bottom Line:

Success requires both geometric insight (recognizing the hypotenuse) and careful algebraic manipulation of radical expressions. The combination of spatial reasoning and algebraic skill makes this a comprehensive test of right triangle knowledge.