Triangle PQR has vertices \(\mathrm{P(1, 0)}\), \(\mathrm{Q(0, 0)}\), and \(\mathrm{R(0, 1)}\) in the coordinate plane. The positive measure of angle...

1. TRANSLATE the problem information

• Given information:
  • Triangle PQR with \(\mathrm{Q(0, 0)}\), \(\mathrm{P(1, 0)}\), \(\mathrm{R(0, 1)}\)
  • Need positive measure of angle PQR from ray QP to ray QR, counterclockwise

• What this tells us:
  • Q is at the origin, so we're measuring from origin
  • Ray QP goes to \(\mathrm{(1, 0)}\) = along positive x-axis
  • Ray QR goes to \(\mathrm{(0, 1)}\) = along positive y-axis

2. VISUALIZE the angle setup

• INFER the basic angle measurement:
  • From positive x-axis to positive y-axis, counterclockwise = \(\frac{\pi}{2}\) radians
  • This is the fundamental angle we're measuring

3. INFER the complete solution approach

• Key insight: "Positive angle measure" means we can include full rotations
• General form: \(\frac{\pi}{2} + 2\pi k\) where k is any non-negative integer
• Need to check which answer choice fits this pattern

4. SIMPLIFY each answer choice

• Check if each equals \(\frac{\pi}{2} + 2\pi k\) for some integer \(\mathrm{k \geq 0}\):

Choice A:

\(\frac{25\pi}{2} = \frac{\pi}{2} + \frac{24\pi}{2}\)

\(= \frac{\pi}{2} + 12\pi\)

\(= \frac{\pi}{2} + 6(2\pi)\)

→ \(\mathrm{k = 6}\) ✓ (integer)

Choice B:

\(\frac{27\pi}{2} = \frac{\pi}{2} + \frac{26\pi}{2}\)

\(= \frac{\pi}{2} + 13\pi\)

→ \(13\pi = 2\pi k\)

→ \(\mathrm{k = 6.5}\) ✗ (not integer)

Choice C:

\(24\pi = \frac{\pi}{2} + 2\pi k\)

→ \(2\pi k = 24\pi - \frac{\pi}{2}\)

\(= \frac{48\pi - \pi}{2}\)

\(= \frac{47\pi}{2}\)

→ \(\mathrm{k = \frac{47}{4} = 11.75}\) ✗ (not integer)

Choice D:

\(\frac{23\pi}{2} = \frac{\pi}{2} + \frac{22\pi}{2}\)

\(= \frac{\pi}{2} + 11\pi\)

→ \(11\pi = 2\pi k\)

→ \(\mathrm{k = 5.5}\) ✗ (not integer)

Choice E:

\(26\pi = \frac{\pi}{2} + 2\pi k\)

→ \(\mathrm{k = \frac{52\pi - \pi}{4\pi}}\)

\(= \frac{51}{4}\)

\(= 12.75\) ✗ (not integer)

5. APPLY CONSTRAINTS to select final answer

• Only choice A gives an integer value for k
• \(\mathrm{k = 6}\) means 6 full counterclockwise rotations plus \(\frac{\pi}{2}\)

Answer: A \(\left(\frac{25\pi}{2}\right)\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students find the basic angle \(\frac{\pi}{2}\) correctly but don't recognize that "positive angle measure" can include multiple rotations. They look for answer choices close to \(\frac{\pi}{2} \approx 1.57\), not realizing the problem allows for coterminal angles.

This leads to confusion since none of the choices are close to \(\frac{\pi}{2}\), causing them to guess or abandon the systematic approach.

Second Most Common Error:

Poor SIMPLIFY execution: Students understand the coterminal angle concept but make algebraic errors when checking if k is an integer. They might incorrectly conclude that other choices work, particularly with decimal/fraction arithmetic involving π.

This may lead them to select Choice B \(\left(\frac{27\pi}{2}\right)\) or Choice D \(\left(\frac{23\pi}{2}\right)\) due to computational mistakes.

The Bottom Line:

This problem tests whether students can bridge basic angle measurement with the concept of coterminal angles, requiring both geometric visualization and algebraic verification skills.