In right triangle PQR, angleQ is a right angle and the length of QR is 84 feet. If sinP =...

1. TRANSLATE the problem information

• Given information:
  • Right triangle PQR with right angle at Q
  • \(\mathrm{QR} = 84\) feet
  • \(\sin \mathrm{P} = \frac{4}{5}\)
  • Find: length of PQ
• What this tells us: We need to identify which sides are opposite, adjacent, and hypotenuse relative to angle P

2. INFER the geometric relationships

• From angle P's perspective in this right triangle:
  • Opposite side = \(\mathrm{QR} = 84\) feet (across from angle P)
  • Adjacent side = \(\mathrm{PQ} = ?\) (next to angle P, what we're finding)
  • Hypotenuse = PR (longest side, opposite the right angle)
• Strategy: Use the given sine value to find the hypotenuse first, then use Pythagorean theorem

3. TRANSLATE the sine definition

• Since \(\sin \mathrm{P} = \frac{\mathrm{opposite}}{\mathrm{hypotenuse}}\):
  \(\sin \mathrm{P} = \frac{\mathrm{QR}}{\mathrm{PR}} = \frac{84}{\mathrm{PR}} = \frac{4}{5}\)

4. SIMPLIFY to find the hypotenuse

• From \(\frac{84}{\mathrm{PR}} = \frac{4}{5}\):
  \(\mathrm{PR} = 84 \times \frac{5}{4} = 105\) feet

5. INFER the next step and apply Pythagorean theorem

• Now we know two sides of the right triangle: \(\mathrm{QR} = 84\) and \(\mathrm{PR} = 105\)
• Use \(\mathrm{a}^2 + \mathrm{b}^2 = \mathrm{c}^2\): \(\mathrm{PQ}^2 + \mathrm{QR}^2 = \mathrm{PR}^2\)

6. SIMPLIFY the Pythagorean equation

• \(\mathrm{PQ}^2 + 84^2 = 105^2\)
• \(\mathrm{PQ}^2 + 7056 = 11025\)
• \(\mathrm{PQ}^2 = 3969\)
• \(\mathrm{PQ} = 63\) feet (use calculator for \(\sqrt{3969}\))

Answer: A (63)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse which side is opposite versus adjacent to angle P, thinking PQ (the side they're looking for) is opposite to angle P instead of QR.

This leads them to set up: \(\sin \mathrm{P} = \frac{\mathrm{PQ}}{\mathrm{PR}} = \frac{4}{5}\), making PQ the unknown in the numerator. They might then incorrectly assume \(\mathrm{PR} = 84\) (confusing it with the given QR value) and calculate \(\mathrm{PQ} = \frac{4}{5} \times 84 = 67.2\), which doesn't match any answer choice exactly, leading to confusion and guessing.

Second Most Common Error:

Poor INFER reasoning: Students recognize the correct sine relationship but try to solve the problem without finding the hypotenuse first. They might attempt to use tangent or cosine relationships without having enough information, or try to set up equations with two unknowns.

This leads them to get stuck partway through and potentially select Choice D (112) by incorrectly calculating \(84 \times \frac{4}{3} = 112\), confusing the reciprocal relationship.

The Bottom Line:

This problem requires careful identification of triangle sides relative to the given angle and strategic thinking about the solution sequence - you must find the hypotenuse before you can find the remaining side.