Lesson 10.3: Inscribed Angles & Sector Measures Mixed Problems

Learning Objectives

• Apply the Inscribed Angle Theorem to determine arc and angle measures.
• Calculate arc length and sector area using central angles.
• Integrate angle and sector relationships in multi-step problem solving.

Topics Covered

• Inscribed Angle Theorem
• Intercepted arcs and angle relationships
• Arc length formula: \( L = \frac{\theta}{360^\circ} \cdot 2\pi r \)
• Sector area formula: \( A = \frac{\theta}{360^\circ} \cdot \pi r^2 \)
• Mixed circle geometry applications

Key Terms

• Inscribed Angle: Angle with its vertex on the circle and sides containing chords.
• Intercepted Arc: Arc cut off by the sides of an inscribed angle.
• Sector: Region bounded by two radii and an arc.
• Arc Length: Distance along the curved edge of a circle.
• Sector Area: Area inside a sector of a circle.

Lesson Examples

Example 1: Finding an Inscribed Angle

An inscribed angle intercepts an arc measuring 120°.

1. Inscribed angle equals half the intercepted arc.
2. \( \text{Angle} = \frac{120^\circ}{2} = 60^\circ \)

Example 2: Finding Arc Length

Find arc length with central angle 90° and radius 6.

1. Use formula \( L = \frac{\theta}{360} \cdot 2\pi r \)
2. \( L = \frac{90}{360} \cdot 2\pi(6) = 3\pi \)

Example 3: Sector Area Using Inscribed Angle Information

An inscribed angle measures 40° in a circle with radius 5.

1. Central angle equals double inscribed angle → 80°
2. Sector area: \( A = \frac{80}{360} \cdot \pi(5^2) \)
3. \( A = \frac{80}{360} \cdot 25\pi = \frac{50\pi}{9} \)

Practice Problems

1. An inscribed angle intercepts a 150° arc. Find the angle.
2. Find the arc length when radius is 8 and central angle is 135°.
3. Find the sector area for a circle with radius 10 and central angle 72°.

Tips & Common Mistakes

• Always verify whether an angle is inscribed or central.
• Inscribed angles are half their intercepted arc, not equal.
• Be careful to use consistent units for radius and arc length.

Summary

Inscribed angles provide a key link between arc measures and sector calculations. By combining angle relationships with circle formulas, students can solve complex geometric problems involving arcs and sectors.

Challenge Problems

1. An inscribed angle intercepts an arc twice as large as another arc in the same circle. Compare the angles.
2. Find the radius of a circle if a 60° sector has area \( 16\pi \).
3. A circle has arc length 12π for a 180° arc. Find the radius and full circumference.