Inscribed Angle Investigation

Project Ref: GEOM-CIRC-01 // Topic: Inscribed vs Central

NAME: _________________________

DATE: _________________________

The Movie Theater Dilemma

If you are watching a movie in a circular theater, does your viewing angle of the screen change depending on where you sit along the back wall? Let's use geometry to find the answer.

Part 1: Data Gathering

Use your geometry software to create a circle with center \(O\). Place two points \(A\) and \(B\) on the circle to create an arc. Place point \(P\) anywhere else on the major arc. Measure the central angle \(\angle AOB\) and the inscribed angle \(\angle APB\).

Trial	Central Angle \(m\angle AOB\)	Inscribed Angle \(m\angle APB\)	Ratio (Central / Inscribed)
1
2
3

Part 2: The Conjecture

Based on your table, what is the relationship between the measure of the central angle and the measure of an inscribed angle that intercepts the same arc?

Drag point \(P\) along the circle. Does the measure of \(\angle APB\) change as long as it stays on the same arc? Why or why not?

The Inscribed Angle Theorem

The measure of an inscribed angle is exactly ___________ the measure of the central angle that intercepts the same arc.

\(m\angle \text{Inscribed} = \frac{1}{2} m\overparen{Arc}\)

Blueprint Practice

1. Solve for \(x\):

Value of \(x\): ___________

2. Solve for \(y\):

Measure of Arc \(y\): ___________

Part 4: Algebraic Applications

In a circle, an inscribed angle is represented by the expression \((3x + 5)^\circ\). The central angle that intercepts the same arc is represented by \((8x - 10)^\circ\).

A. Set up an equation to find \(x\):

B. Solve for \(x\):

C. What is the measure of the inscribed angle?

Reflection: Why must the vertex of the inscribed angle be ON the circle for this theorem to work? What happens if the vertex moves inside or outside the circle?

Circle Secrets

Inscribed Angle Discovery

Angles Subtending Same Arc

Thales's Theorem and Right Triangles

Properties of Cyclic Quadrilaterals

Tangent-Radius Relationships

Angle Discovery Worksheet

Angle Discovery Slides

Angle Discovery Key

Arc Sharing Slides

Bowtie Angles Worksheet

Bowtie Angles Key

Semicircle Secrets Slides

Thales's Theorem Worksheet

Thales's Theorem Key

Cyclic Quad Slides

Cyclic Quad Worksheet

Cyclic Quad Key

Tangent Trail Slides

Tangent Task Worksheet

Tangent Task Key

Similar Sequences

Function Frontiers

Jessica Waters

Money Math Mastery

Radical Relationships

Calculus Foundations

Urban Concurrency Planning

Transformation Surgeons

Probability Patterns

Linear Landscapes

Power Root Mastery

Human Calculator Series

Sigma Summit

Power of Growth

Pattern Detectives

Recursive Sequence Mastery

Exponent Mastery

Linear Rescue

Slope and Symmetry

Shopping Math Mastery

Sequence Science Unit

Intersection Insights