9.3: Arcs in Circles
What if the Ferris wheel below had equally spaced seats, such that the central angle were \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.
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CK-12 Foundation: Chapter9ArcsinCirclesA
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 \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 \begin{align*}\widehat{BC}\end{align*} and \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 \begin{align*}\overline{BC}\end{align*} and \begin{align*}\widehat{BC}\end{align*}.
If \begin{align*}D\end{align*} was not on the circle, we would not be able to tell the difference between \begin{align*}\widehat{BC}\end{align*} and \begin{align*}\widehat{BDC}\end{align*}. There are \begin{align*}360^\circ\end{align*} in a circle, where a semicircle is half of a circle, or \begin{align*}180^\circ\end{align*}. \begin{align*}m \angle EFG = 180^\circ\end{align*}, because it is a straight angle, so \begin{align*}m \widehat{EHG}= 180^\circ\end{align*} and \begin{align*}m \widehat{EJG} = 180^\circ\end{align*}.
- Semicircle: An arc that measures \begin{align*}180^\circ\end{align*}.
- Minor Arc: An arc that is less than \begin{align*}180^\circ\end{align*}.
- Major Arc: An arc that is greater than \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 \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 “\begin{align*}m\end{align*}” in from of the label.
Example A
Find \begin{align*}m\widehat{AB}\end{align*} and \begin{align*}m\widehat{ADB}\end{align*} in \begin{align*}\bigodot C\end{align*}.
\begin{align*}m\widehat{AB}= m\angle{ACB}\end{align*}. So, \begin{align*}m\widehat{AB}= 102^\circ\end{align*}.
\begin{align*}m\widehat{ADB}=360^\circ - m\widehat{AB}=360^\circ-102^\circ=258^\circ\end{align*}
Example B
Find the measures of the minor arcs in \begin{align*}\bigodot{A}\end{align*}. \begin{align*}\overline{EB}\end{align*} is a diameter.
Because \begin{align*}\overline{EB}\end{align*} is a diameter, \begin{align*}m\angle EAB=180^\circ\end{align*}. Each arc has the same measure as its corresponding central angle.
\begin{align*}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\end{align*}
Example C
Find the measures of the indicated arcs in \begin{align*}\bigodot A\end{align*}. \begin{align*}\overline{EB}\end{align*} is a diameter.
a) \begin{align*}m\widehat{FED}\end{align*}
b) \begin{align*}m\widehat{CDF}\end{align*}
c) \begin{align*}m\widehat{DFC}\end{align*}
Use the Arc Addition Postulate.
a) \begin{align*}m\widehat{FED} = m\widehat{FE} +m\widehat{ED} = 120^\circ+38^\circ=158^\circ\end{align*}
b) \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) \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.
CK-12 Foundation: Chapter9ArcsinCirclesB
Concept Problem Revisited
Because the seats are \begin{align*}20^\circ\end{align*} apart, there will be \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 \begin{align*}x\end{align*} and then multiply it by 2.
\begin{align*}\sin 10^\circ &= \frac{x}{25}\\ x = 25 \sin 10^\circ &= 4.3 \ ft.\end{align*}
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 \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 \begin{align*}180^\circ\end{align*}. A major arc is an arc that is greater than \begin{align*}180^\circ\end{align*}.
Guided Practice
1. List the congruent arcs in \begin{align*}\bigodot C\end{align*} below. \begin{align*}\overline{AB}\end{align*} and \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 \begin{align*}x\end{align*} for \begin{align*}\bigodot C\end{align*} below.
Answers:
1. \begin{align*}\angle ACD \cong \angle ECB\end{align*} because they are vertical angles. \begin{align*}\angle DCB \cong \angle ACE\end{align*} because they are also vertical angles.
\begin{align*}\widehat{AD} \cong \widehat{EB}\end{align*} and \begin{align*}\widehat{AE} \cong \widehat{DB}\end{align*}
2. a) \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 \begin{align*}360^\circ\end{align*} because they make a full circle.
\begin{align*}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\end{align*}
Practice
Determine if the arcs below are a minor arc, major arc, or semicircle of \begin{align*}\bigodot G\end{align*}. \begin{align*}\overline{EB}\end{align*} is a diameter.
- \begin{align*}\widehat{AB}\end{align*}
- \begin{align*}\widehat{ABD}\end{align*}
- \begin{align*}\widehat{BCE}\end{align*}
- \begin{align*}\widehat{CAE}\end{align*}
- \begin{align*}\widehat{ABC}\end{align*}
- \begin{align*}\widehat{EAB}\end{align*}
- Are there any congruent arcs? If so, list them.
- If \begin{align*}m \widehat{BC}=48^\circ\end{align*}, find \begin{align*}m \widehat{CD}\end{align*}.
- Using #8, find \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 \begin{align*}\bigodot A\end{align*}. \begin{align*}\overline{DG}\end{align*} is a diameter.
- \begin{align*}\widehat{DE}\end{align*}
- \begin{align*}\widehat{DC}\end{align*}
- \begin{align*}\angle GAB\end{align*}
- \begin{align*}\widehat{FG}\end{align*}
- \begin{align*}\widehat{EDB}\end{align*}
- \begin{align*}\angle EAB\end{align*}
- \begin{align*}\widehat{DCF}\end{align*}
- \begin{align*}\widehat{DBE}\end{align*}
Algebra Connection Find the measure of \begin{align*}x\end{align*} in \begin{align*}\bigodot P\end{align*}.
- What can you conclude about \begin{align*}\bigodot A\end{align*} and \begin{align*}\bigodot B\end{align*}?
arc
A single section of the circle, that describes a particular angle.central angle
An angle formed by two radii and whose vertex is at the center of the circle.major arc
An arc that is greater than .minor arc
An arc that is less than .semicircle
An arc that measures .Arc Addition Postulate
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 is the measure of the distance across the center of a circle. The diameter is equal to twice the measure of the radius.Image Attributions
Here you'll learn the properties of arcs and central angles of circles and how to apply them.