Math Problem Statement
Match each indicated measure in the first column for circle O. Find arc BC, angle B, angle C, and arc AB given that angle BAC = 48° and angle AOC = 110°.
Solution
To solve this geometry problem, let's analyze the circle diagram step by step to match each measure:
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Key Observations:
- The figure is a circle with center and points on the circle.
- Given measures are and .
- The answers to drag are: .
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Understanding Central and Inscribed Angles:
- A central angle is measured at the center of the circle, and its measure is equal to the arc it intercepts.
- An inscribed angle is measured on the circle and equals half the intercepted arc.
Step-by-Step Solution:
1. Find (arc BC):
- is a central angle, and it intercepts arc .
Thus:
2. Find (inscribed angle at B):
- Inscribed angles intercept arcs, and they are half the intercepted arc.
Here, intercepts arc , which is complementary to arc on the circle.
The remaining arc is:
Thus, the inscribed angle is:
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Math Problem Analysis
Mathematical Concepts
Geometry
Circle Theorems
Angles and Arcs
Formulas
Central Angle = Measure of Intercepted Arc
Inscribed Angle = 1/2 × Intercepted Arc
Sum of Arcs in a Circle = 360°
Theorems
Central Angle Theorem
Inscribed Angle Theorem
Suitable Grade Level
Grades 9-10
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