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:

  1. Key Observations:

    • The figure is a circle with center OO and points A,B,CA, B, C on the circle.
    • Given measures are BAC=48\angle BAC = 48^\circ and AOC=110\angle AOC = 110^\circ.
    • The answers to drag are: 96,77, and 15496^\circ, 77^\circ, \text{ and } 154^\circ.
  2. 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 mBC^m\widehat{BC} (arc BC):

  • AOC=110\angle AOC = 110^\circ is a central angle, and it intercepts arc BCBC.
    Thus:
    mBC^=110.m\widehat{BC} = 110^\circ.

2. Find mBm\angle B (inscribed angle at B):

  • Inscribed angles intercept arcs, and they are half the intercepted arc.
    Here, B\angle B intercepts arc ACAC, which is complementary to arc BCBC on the circle.
    The remaining arc ACAC is:
    mAC^=360mBC^=360110=250.m\widehat{AC} = 360^\circ - m\widehat{BC} = 360^\circ - 110^\circ = 250^\circ. Thus, the inscribed angle B\angle B is:
    mB=12×250=125.m\angle B = \frac{1}{2} \times 250^\circ = 125^\circ.

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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