Central Angle Math Example 2

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Example 2

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In a circle, three central angles divide the circle into three arcs with measures 3x°3x°, 5x°5x°, and 4x°4x°. Find each arc and central angle.

Solution

  1. 1
    Step 1: All arcs of a circle sum to 360°360° (a full circle). So 3x+5x+4x=3603x + 5x + 4x = 360.
  2. 2
    Step 2: 12x=36012x = 360, giving x=30x = 30.
  3. 3
    Step 3: The three arcs are: 3(30)=90°3(30) = 90°, 5(30)=150°5(30) = 150°, 4(30)=120°4(30) = 120°.
  4. 4
    Step 4: By the Central Angle Theorem, each central angle equals its intercepted arc: 90°90°, 150°150°, 120°120°.

Answer

Arcs (and corresponding central angles): 90°90°, 150°150°, 120°120°.
The sum of all arcs in a circle is 360°. Setting up this equation with the algebraic arc expressions and solving for xx gives each arc measure. Since central angles equal their intercepted arcs, the central angles have the same measures.

About Central Angle

An angle whose vertex is at the center of a circle, with its two rays intersecting the circle at two points. Its measure equals the measure of the intercepted arc.

Learn more about Central Angle →

More Central Angle Examples

Example 1 easy

A central angle in a circle intercepts an arc of [formula]. What is the measure of the central angle

Example 3 easy

A circle is divided into 6 equal sectors. What is the central angle of each sector?

Example 4 hard

In circle [formula], central angle [formula] and central angle [formula]. Find arc [formula] going t

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