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# 9.3: Arcs in Circles

Difficulty Level: At Grade Created by: CK-12
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What if the Ferris wheel below had equally spaced seats, such that the central angle were 20\begin{align*}20^\circ\end{align*}. How many seats are there? Why do you think it is important to have equally spaced seats on a Ferris wheel?

If the radius of this Ferris wheel is 25 ft., how far apart are two adjacent seats? Round your answer to the nearest tenth. The shortest distance between two points is a straight line. After completing this Concept, you'll be able to answer questions like these.

### Guidance

A central angle is the angle formed by two radii of the circle with its vertex at the center of the circle. In the picture below, the central angle would be BAC\begin{align*}\angle BAC\end{align*}. Every central angle divides a circle into two arcs (an arc is a section of the circle). In this case the arcs are BCˆ\begin{align*}\widehat{BC}\end{align*} and BDCˆ\begin{align*}\widehat{BDC}\end{align*}. Notice the arc above the letters. To label an arc, always use this curve above the letters. Do not confuse BC¯¯¯¯¯\begin{align*}\overline{BC}\end{align*} and BCˆ\begin{align*}\widehat{BC}\end{align*}.

If D\begin{align*}D\end{align*} was not on the circle, we would not be able to tell the difference between BCˆ\begin{align*}\widehat{BC}\end{align*} and BDCˆ\begin{align*}\widehat{BDC}\end{align*}. There are 360\begin{align*}360^\circ\end{align*} in a circle, where a semicircle is half of a circle, or 180\begin{align*}180^\circ\end{align*}. mEFG=180\begin{align*}m \angle EFG = 180^\circ\end{align*}, because it is a straight angle, so mEHGˆ=180\begin{align*}m \widehat{EHG}= 180^\circ\end{align*} and mEJGˆ=180\begin{align*}m \widehat{EJG} = 180^\circ\end{align*}.

• Semicircle: An arc that measures 180\begin{align*}180^\circ\end{align*}.
• Minor Arc: An arc that is less than 180\begin{align*}180^\circ\end{align*}.
• Major Arc: An arc that is greater than 180\begin{align*}180^\circ\end{align*}. Always use 3 letters to label a major arc.

Two arcs are congruent if their central angles are congruent. The measure of the arc formed by two adjacent arcs is the sum of the measures of the two arcs (Arc Addition Postulate). An arc can be measured in degrees or in a linear measure (cm, ft, etc.). In this chapter we will use degree measure. The measure of the minor arc is the same as the measure of the central angle that corresponds to it. The measure of the major arc equals to 360\begin{align*}360^\circ\end{align*} minus the measure of the minor arc. In order to prevent confusion, major arcs are always named with three letters; the letters that denote the endpoints of the arc and any other point on the major arc. When referring to the measure of an arc, always place an “m\begin{align*}m\end{align*}” in from of the label.

#### Example A

Find mABˆ\begin{align*}m\widehat{AB}\end{align*} and mADBˆ\begin{align*}m\widehat{ADB}\end{align*} in C\begin{align*}\bigodot C\end{align*}.

mABˆ=mACB\begin{align*}m\widehat{AB}= m\angle{ACB}\end{align*}. So, mABˆ=102\begin{align*}m\widehat{AB}= 102^\circ\end{align*}.

#### Example B

Find the measures of the minor arcs in A\begin{align*}\bigodot{A}\end{align*}. EB¯¯¯¯¯\begin{align*}\overline{EB}\end{align*} is a diameter.

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

#### Example C

Find the measures of the indicated arcs in A\begin{align*}\bigodot A\end{align*}. EB¯¯¯¯¯\begin{align*}\overline{EB}\end{align*} is a diameter.

a) mFEDˆ\begin{align*}m\widehat{FED}\end{align*}

b) mCDFˆ\begin{align*}m\widehat{CDF}\end{align*}

c) mDFCˆ\begin{align*}m\widehat{DFC}\end{align*}

a) mFEDˆ=mFEˆ+mEDˆ=120+38=158\begin{align*}m\widehat{FED} = m\widehat{FE} +m\widehat{ED} = 120^\circ+38^\circ=158^\circ\end{align*}

b) mCDFˆ=mCDˆ+mDEˆ+mEFˆ=90+38+120=248\begin{align*}m\widehat{CDF} = m\widehat{CD} + m \widehat{DE} + m \widehat{EF} = 90^\circ + 38^\circ + 120^\circ = 248^\circ\end{align*}

c) mDFCˆ=mEDˆ+mEFˆ+mFBˆ+mBCˆ=38+120+60+52=270\begin{align*}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\end{align*}

Watch this video for help with the Examples above.

#### Concept Problem Revisited

Because the seats are 20\begin{align*}20^\circ\end{align*} apart, there will be 36020=18\begin{align*}\frac{360^\circ}{20^\circ}=18\end{align*} seats. It is important to have the seats evenly spaced for balance. To determine how far apart the adjacent seats are, use the triangle to the right. We will need to use sine to find x\begin{align*}x\end{align*} and then multiply it by 2.

sin10x=25sin10=x25=4.3 ft.

The total distance apart is 8.6 feet.

### Vocabulary

A circle is the set of all points that are the same distance away from a specific point, called the center. An arc is a section of the circle. A semicircle is an arc that measures 180\begin{align*}180^\circ\end{align*}. 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\begin{align*}180^\circ\end{align*}. A major arc is an arc that is greater than 180\begin{align*}180^\circ\end{align*}.

### Guided Practice

1. List the congruent arcs in C\begin{align*}\bigodot C\end{align*} below. AB¯¯¯¯¯\begin{align*}\overline{AB}\end{align*} and DE¯¯¯¯¯\begin{align*}\overline{DE}\end{align*} are diameters.

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

a)

b)

3. Find the value of x\begin{align*}x\end{align*} for C\begin{align*}\bigodot C\end{align*} below.

1. ACDECB\begin{align*}\angle ACD \cong \angle ECB\end{align*} because they are vertical angles. DCBACE\begin{align*}\angle DCB \cong \angle ACE\end{align*} because they are also vertical angles.

ADˆEBˆ\begin{align*}\widehat{AD} \cong \widehat{EB}\end{align*} and AEˆDBˆ\begin{align*}\widehat{AE} \cong \widehat{DB}\end{align*}

2. a) ADˆBCˆ\begin{align*}\widehat{AD} \cong \widehat{BC}\end{align*} 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\begin{align*}360^\circ\end{align*} because they make a full circle.

### Practice

Determine if the arcs below are a minor arc, major arc, or semicircle of G\begin{align*}\bigodot G\end{align*}. EB¯¯¯¯¯\begin{align*}\overline{EB}\end{align*} is a diameter.

1. ABˆ\begin{align*}\widehat{AB}\end{align*}
2. ABDˆ\begin{align*}\widehat{ABD}\end{align*}
3. BCEˆ\begin{align*}\widehat{BCE}\end{align*}
4. CAEˆ\begin{align*}\widehat{CAE}\end{align*}
5. ABCˆ\begin{align*}\widehat{ABC}\end{align*}
6. EABˆ\begin{align*}\widehat{EAB}\end{align*}
7. Are there any congruent arcs? If so, list them.
8. If mBCˆ=48\begin{align*}m \widehat{BC}=48^\circ\end{align*}, find mCDˆ\begin{align*}m \widehat{CD}\end{align*}.
9. Using #8, find mCAEˆ\begin{align*}m \widehat{CAE}\end{align*}.

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

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

1. DEˆ\begin{align*}\widehat{DE}\end{align*}
2. DCˆ\begin{align*}\widehat{DC}\end{align*}
3. GAB\begin{align*}\angle GAB\end{align*}
4. FGˆ\begin{align*}\widehat{FG}\end{align*}
5. EDBˆ\begin{align*}\widehat{EDB}\end{align*}
6. EAB\begin{align*}\angle EAB\end{align*}
7. DCFˆ\begin{align*}\widehat{DCF}\end{align*}
8. DBEˆ\begin{align*}\widehat{DBE}\end{align*}

Algebra Connection Find the measure of x\begin{align*}x\end{align*} in P\begin{align*}\bigodot P\end{align*}.

1. What can you conclude about A\begin{align*}\bigodot A\end{align*} and B\begin{align*}\bigodot B\end{align*}?

### Vocabulary Language: English Spanish

arc

arc

A single section of the circle, that describes a particular angle.
central angle

central angle

An angle formed by two radii and whose vertex is at the center of the circle.
major arc

major arc

An arc that is greater than $180^\circ$.
minor arc

minor arc

An arc that is less than $180^\circ$.
semicircle

semicircle

An arc that measures $180^\circ$.

Arc addition postulate states that the measure of the arc formed by two adjacent arcs is the sum of the measures of the two arcs.
Diameter

Diameter

Diameter is the measure of the distance across the center of a circle. The diameter is equal to twice the measure of the radius.

Jul 17, 2012

Feb 26, 2015