What if the Ferris wheel below had equally spaced seats, such that the central angle were
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.
Watch This
CK12 Foundation: Chapter9ArcsinCirclesA
Learn more about central angles by watching the video at this link.
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
If

Semicircle: An arc that measures
180∘ .

Minor Arc: An arc that is less than
180∘ .

Major Arc: An arc that is greater than
180∘ . 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
Example A
Find
Example B
Find the measures of the minor arcs in
Because
Example C
Find the measures of the indicated arcs in
a)
b)
c)
Use the Arc Addition Postulate.
a)
b)
c)
Watch this video for help with the Examples above.
CK12 Foundation: Chapter9ArcsinCirclesB
Concept Problem Revisited
Because the seats are
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
Guided Practice
1. List the congruent arcs in
2. Are the blue arcs congruent? Explain why or why not.
a)
b)
3. Find the value of
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*}
Interactive Practice
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*}?