In triangle PQR, angle R is a right angle. The measure of angle P is \((3\mathrm{t} + 12)^\circ\), and the...

1. TRANSLATE the problem information

• Given information:
  • Triangle PQR has a right angle at R (so \(\mathrm{angle\; R = 90°}\))
  • Angle P measures \(\mathrm{(3t + 12)}\) degrees
  • Angle Q measures \(\mathrm{(2t + 18)}\) degrees
  • We need to find the value of t

2. INFER the key relationship

• Since angle R is 90°, angles P and Q are the two acute angles in this right triangle
• Key insight: In any right triangle, the two acute angles are complementary (they add up to 90°)
• This means: \(\mathrm{Angle\; P + Angle\; Q = 90°}\)

3. TRANSLATE the relationship into an equation

• Set up the equation: \(\mathrm{(3t + 12) + (2t + 18) = 90}\)

4. SIMPLIFY to solve for t

• Combine like terms: \(\mathrm{3t + 2t + 12 + 18 = 90}\)
• This gives us: \(\mathrm{5t + 30 = 90}\)
• Subtract 30 from both sides: \(\mathrm{5t = 60}\)
• Divide by 5: \(\mathrm{t = 12}\)

5. Check the solution

• If \(\mathrm{t = 12}\): \(\mathrm{Angle\; P = 3(12) + 12 = 48°}\), \(\mathrm{Angle\; Q = 2(12) + 18 = 42°}\)
• Verification: \(\mathrm{48° + 42° = 90°}\) ✓

Answer: C. 12

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students forget that the two acute angles in a right triangle are complementary and instead try to use the full triangle angle sum (180°).

They set up: \(\mathrm{(3t + 12) + (2t + 18) + 90 = 180}\)

This leads to: \(\mathrm{5t + 120 = 180}\), so \(\mathrm{5t = 60}\), and \(\mathrm{t = 12}\).

While this actually gives the same correct answer, it's a less efficient approach that shows they missed the more direct complementary angle relationship.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the complementary relationship but make arithmetic errors when combining like terms or solving the equation.

For example, they might incorrectly combine \(\mathrm{3t + 2t = 6t}\) instead of \(\mathrm{5t}\), or miscalculate \(\mathrm{12 + 18 = 28}\) instead of \(\mathrm{30}\).

This leads to incorrect values of t and may cause them to select one of the wrong answer choices.

The Bottom Line:

The key insight is recognizing that right triangles have a special property - their two acute angles always sum to exactly 90°. Students who miss this relationship often still solve the problem correctly but use a more complicated approach, while those who make algebraic errors will select incorrect answers.