Central Angle Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

hard
In circle OO, central angle AOB=80°AOB = 80° and central angle BOC=110°BOC = 110°. Find arc ACAC going the short way (not through BB), and the reflex arc ACAC going through BB.

Solution

  1. 1
    Step 1: Arc AB=80°AB = 80° and arc BC=110°BC = 110° (central angles equal intercepted arcs).
  2. 2
    Step 2: Reflex arc ACAC (going through BB) =arc AB+arc BC=80°+110°=190°= \text{arc } AB + \text{arc } BC = 80° + 110° = 190°.
  3. 3
    Step 3: Minor arc ACAC (not through BB) =360°−190°=170°= 360° - 190° = 170°.
  4. 4
    Step 4: The central angle AOCAOC (for the minor arc) =170°= 170°.

Answer

Minor arc AC=170°AC = 170°; Reflex arc ACAC (through BB) =190°= 190°.
When multiple central angles are given, add the corresponding arcs to find a composite arc. The full circle is 360°, so the remaining arc is found by subtracting the known arc from 360°. A reflex arc is one greater than 180° (more than a semicircle).

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 2 medium

In a circle, three central angles divide the circle into three arcs with measures [formula], [formul

Example 3 easy

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

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