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Arcs in Circles

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Arcs in Circles

What if a circle were divided into pieces by various radii? How could you find the measures of the arcs formed by these radii? After completing this Concept, you'll be able to use central angles to solve problems like this one.

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Arcs in Circles CK-12

Guidance

A circle has 360^\circ . An arc is a section of the circle. A semicircle is an arc that measures 180^\circ .

\widehat{EHG} \ \text{and} \ \widehat{EJG} \ \text{are semicircles}

A central angle is the angle formed by two radii with its vertex at the center of the circle. A minor arc is an arc that is less than 180^\circ . A major arc is an arc that is greater than 180^\circ . Always use 3 letters to label a major arc.

The central angle is \angle BAC . The minor arc is \widehat{BC} . The major arc is \widehat{BDC} .

An arc can be measured in degrees or in a linear measure (cm, ft, etc.). In this concept we will use degree measure. The measure of a minor arc is the same as the measure of the central angle that corresponds to it . The measure of a major arc is 360^\circ minus the measure of the corresponding minor arc. The measure of the arc formed by two adjacent arcs is the sum of the measures of the two arcs ( Arc Addition Postulate ).

m\widehat{AD} + m\widehat{DB} = m\widehat{ADB}

Example A

Find m\widehat{AB} and m\widehat{ADB} in \bigodot C .

m\widehat{AB}= m\angle{ACB} . So, m\widehat{AB}= 102^\circ .

m\widehat{ADB}=360^\circ - m\widehat{AB}=360^\circ-102^\circ=258^\circ

Example B

Find the measures of the minor arcs in \bigodot{A} . \overline{EB} is a diameter.

Because \overline{EB} is a diameter, m\angle EAB=180^\circ . Each arc has the same measure as its corresponding central angle.

m \widehat{BF} & = m \angle FAB = 60^\circ\\m\widehat{EF} & = m \angle EAF = 120^\circ \rightarrow 180^\circ - 60^\circ\\m\widehat{ED} & = m \angle EAD = 38^\circ \ \rightarrow 180^\circ - 90^\circ - 52^\circ\\m\widehat{DC} & = m \angle DAC = 90^\circ\\m\widehat{BC} & = m \angle BAC = 52^\circ

Example C

Find the measures of the indicated arcs in \bigodot A . \overline{EB} is a diameter.

a) m\widehat{FED}

b) m\widehat{CDF}

c) m\widehat{DFC}

Use the Arc Addition Postulate.

a) m\widehat{FED} = m\widehat{FE} +m\widehat{ED} = 120^\circ+38^\circ=158^\circ

b) m\widehat{CDF} = m\widehat{CD} + m \widehat{DE} + m \widehat{EF} = 90^\circ + 38^\circ + 120^\circ = 248^\circ

c) m \widehat{DFC} = m\widehat{ED} + m\widehat{EF} + m\widehat{FB} + m\widehat{BC} = 38^\circ + 120^\circ + 60^\circ + 52^\circ = 270^\circ

Arcs in Circles CK-12

Guided Practice

1. List the congruent arcs in \bigodot C below. \overline{AB} and \overline{DE} are diameters.

2. Are the blue arcs congruent? Explain why or why not.

a)

b)

3. Find the value of x for \bigodot C below.

Answers:

1. \angle ACD \cong \angle ECB because they are vertical angles. \angle DCB \cong \angle ACE because they are also vertical angles.

\widehat{AD} \cong \widehat{EB} and \widehat{AE} \cong \widehat{DB}

2. a) \widehat{AD} \cong \widehat{BC} because they have the same central angle measure and are in the same circle.

b) The two arcs have the same measure, but are not congruent because the circles have different radii.

3. The sum of the measure of the arcs is 360^\circ because they make a full circle.

m \widehat{AB} + m \widehat{AD} + m \widehat{DB} & = 360^\circ\\(4x+15)^\circ+92^\circ+(6x+3)^\circ&=360^\circ\\10x+110^\circ&=360^\circ\\10x&=250\\x&=25

Practice

Determine whether the arcs below are a minor arc, major arc, or semicircle of \bigodot G . \overline{EB} is a diameter.

  1. \widehat{AB}
  2. \widehat{ABD}
  3. \widehat{BCE}
  4. \widehat{CAE}
  5. \widehat{ABC}
  6. \widehat{EAB}
  7. Are there any congruent arcs? If so, list them.
  8. If m\widehat{BC} = 48^\circ , find m\widehat{CD} .
  9. Using #8, find m \widehat{CAE} .

Find the measure of the minor arc and the major arc in each circle below.

Determine if the blue arcs are congruent. If so, state why.

Find the measure of the indicated arcs or central angles in \bigodot A . \overline{DG} is a diameter.

  1. \widehat{DE}
  2. \widehat{DC}
  3. \widehat{GAB}
  4. \widehat{FG}
  5. \widehat{EDB}
  6. \widehat{EAB}
  7. \widehat{DCF}
  8. \widehat{DBE}

Find the measure of x in \bigodot P .

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