Point F lies on a unit circle in the xy-plane and has coordinates \((1, 0)\). Point G is the center...

1. TRANSLATE the problem information

• Given information:
  • Unit circle \(\mathrm{(radius = 1)}\) centered at \(\mathrm{G(0, 0)}\)
  • Point F at \(\mathrm{(1, 0)}\) on the circle
  • Point H at \(\mathrm{(-1, y)}\) on the circle
  • Need positive measure of angle FGH in radians

2. INFER that H's coordinates must satisfy the unit circle

• Since H is on a unit circle centered at origin:
  • Distance from G to H must equal 1
  • This gives us: \(\sqrt{(-1)^2 + y^2} = 1\)

3. SIMPLIFY to find y

• \(\sqrt{1 + y^2} = 1\)
• Square both sides: \(1 + y^2 = 1\)
• Therefore: \(y^2 = 0\), so \(y = 0\)
• Point H is at \(\mathrm{(-1, 0)}\)

4. VISUALIZE the angle setup

• F at \(\mathrm{(1, 0)}\): positive x-axis
• H at \(\mathrm{(-1, 0)}\): negative x-axis
• G at origin: vertex of angle
• Basic angle from positive to negative x-axis = \(\pi\) radians

5. INFER the general form of positive angle measures

• The basic angle is \(\pi\), but we can add full rotations
• General form: \(\pi + 2\pi k\) where \(k \geq 0\)
• This gives us all odd multiples of \(\pi\): \(\pi, 3\pi, 5\pi, 7\pi, ..., 25\pi, ...\)

6. APPLY CONSTRAINTS to select the correct answer

• Check which choice is an odd multiple of \(\pi\):
  1. \(27\pi/2 = 13.5\pi\) ✗
  2. \(29\pi/2 = 14.5\pi\) ✗
  3. \(24\pi\) ✗ (even multiple)
  4. \(25\pi\) ✓ (odd multiple: \(\pi + 24\pi\))

Answer: D. \(25\pi\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that H's position must satisfy the unit circle constraint. They might assume y could be any value, leading to confusion about the angle's location. Without finding that H is at \(\mathrm{(-1, 0)}\), they can't determine the correct angle relationship. This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor TRANSLATE reasoning: Students interpret "positive measure" as meaning only the basic angle \(\pi\), not recognizing that angles can have coterminal measures. They might calculate the basic angle correctly but then look for \(\pi\) among the choices. Since \(\pi\) isn't listed, this causes them to get stuck and guess.

The Bottom Line:

This problem tests both coordinate geometry skills (finding points on circles) and angle measurement concepts (coterminal angles). Students need to work systematically through the constraint that H lies on the unit circle, then understand that angle measures aren't unique - they can include multiple rotations.