8.3: Arcs, Semi-Circles, and Central Angles
Learning Objectives
- Measure central angles.
- Measure arcs of circles.
- Find relationships between minor arcs, semicircles, and major arcs.
Arc, Central Angle
In a circle, the central angle is formed by two radii of the circle with its vertex at the center of the circle. An arc is a section of the circle.
A circle’s central angle is an angle with a vertex at the _____________________ of the circle.
Each side of a central angle is also a _______________________ of the circle.
An ________________ is a section of the outer edge of the circle.
Minor and Major Arcs, Semicircle
A semicircle is half of a circle.
A major arc is longer than a semicircle and a minor arc is shorter than a semicircle.
A ______________________________ is half of a circle.
A __________________ arc is longer than a semicircle and a __________________ arc is shorter than a semicircle.
An arc can be measured in degrees or in a linear measure (cm, ft, etc.). In this lesson we will concentrate on degree measure.
The measure of the minor arc is the same as the measure of the central angle that corresponds to it.
This means that the measure of the arc in between the sides of the central angle (from point \begin{align*}A\end{align*} to point \begin{align*}B\end{align*} below) is the same degree measure as the central angle, like in the picture below:
image from http://www.regentsprep.org/Regents/math/geometry/GP15/CircleArcs.htm
The measure of the major arc is equal to \begin{align*}360^\circ\end{align*} minus the measure of the minor arc.
In the example above, the larger arc from point \begin{align*}A\end{align*} to point \begin{align*}B\end{align*} (in a counter-clockwise direction) is \begin{align*}360^\circ - 85^\circ = 275^\circ\end{align*}. Minor arcs are named with two letters—the letters that denote the endpoints of the arc. Below, the minor arc corresponding to the central angle \begin{align*}\angle AOC\end{align*} is called \begin{align*}\widehat{AC}\end{align*}.
image from http://www.icoachmath.com/SiteMap/MajorArc.html
In order to prevent confusion, semicircles and major arcs are named with three letters—the letters that denote the endpoints of the arc and any other point on the major arc. In the figure above, the major arc corresponding to the central angle \begin{align*}\angle AOC\end{align*} is called \begin{align*}\widehat{ABC}\end{align*} because point \begin{align*}B\end{align*} is on the major arc.
Minor arcs are named with ________ letters, which represent the __________________ of the arc.
Major arcs are named with ________ letters, which represent the endpoints of the arc and a _________________ on the major arc.
Congruent Arcs
Two arcs that correspond to congruent central angles will also be congruent.
In the figure below, \begin{align*}\angle AOC \cong \angle BOD\end{align*} because they are vertical angles. This also means that \begin{align*}\widehat{AC} \cong \widehat{DB}\end{align*}:
If central angles are congruent, their corresponding arcs are also __________________.
Arc Addition Postulate
The measure of the arc formed by two adjacent arcs (or two arcs next to each other) is the sum of the measures of the two arcs.
In other words, \begin{align*}m \widehat{RQ} = m \widehat{RP} + m \widehat{PQ}\end{align*}:
Reading Check:
1. True/False: A central angle in a circle creates two arcs along its circumference: the minor arc is the smaller arc and the major arc is the larger arc.
2. True/False: If two central angles in a circle are congruent, then their corresponding arcs are supplementary.
3. For question #2 above, give an example that proves why this statement is either true or false. Explain your reasoning in words.
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Chord, Diameter, Secant
A chord is defined as a line segment starting at one point on the circle and ending at another point on the circle.
A chord that goes through the center of the circle is called the diameter of the circle. Notice that the diameter is twice as long as the radius of the circle.
A secant is a line that cuts through the circle and continues infinitely in both directions.
- A line segment from one point on a circle to another point on the circle is called a _____________________.
- When a chord goes through the center of a circle, it is a ___________________.
- A chord that is a line extending in both directions is a _____________________.
Congruent Chords Have Congruent Minor Arcs
In the same circle or congruent circles, congruent chords have congruent minor arcs.
Proof. Draw the diagram above, where the chords \begin{align*}\overline{DB}\end{align*} and \begin{align*}\overline{AC}\end{align*} are congruent. Construct \begin{align*}\Delta DOB\end{align*} and \begin{align*}\Delta AOC\end{align*} by drawing the 4 radii from the center \begin{align*}O\end{align*} to points \begin{align*}A, B, C,\end{align*} and \begin{align*}D\end{align*}.
Then, \begin{align*}\Delta AOC \cong \Delta BOD\end{align*} by the SSS Postulate.
This means that central angles \begin{align*}\angle AOC \cong \angle BOD\end{align*} (by CPCTC), which leads to the conclusion that \begin{align*}\widehat{AC} \cong \widehat{DB}\end{align*}.
Congruent Minor Arcs Have Congruent Chords and Congruent Central Angles
In the same circle or congruent circles, congruent minor arcs have congruent chords and congruent central angles.
Proof. Draw the following diagram, in which \begin{align*}\widehat{AC} \cong \widehat{DB}\end{align*}. In the diagram, \begin{align*}\overline{DO}, \overline{OB}, \overline{AO}\end{align*}, and \begin{align*}\overline{OC}\end{align*} are each a radius of the circle.
Since \begin{align*}\widehat{AC} \cong \widehat{DB}\end{align*}, this means that the corresponding central angles are also congruent:
\begin{align*}\angle AOC \cong \angle BOD\end{align*}
Therefore, \begin{align*}\Delta AOC \cong \Delta BOD\end{align*} by the SAS Postulate.
We conclude that \begin{align*}\overline{DB} \cong \overline{AC}\end{align*}.
Here are some examples in which we apply the concepts and theorems we discussed in this lesson.
Example 1
Find the measure of each arc:
A. \begin{align*}m \widehat{ML}\end{align*}
B. \begin{align*}m \widehat{PM}\end{align*}
C. \begin{align*}m \widehat{LMQ}\end{align*}
Solutions:
A. \begin{align*}m \widehat{ML}\end{align*} is equal to \begin{align*}m \angle LOM\end{align*} (the central angle) \begin{align*}= 60^\circ\end{align*}
B. \begin{align*}m \widehat{PM}\end{align*} is equal to \begin{align*}m \angle POM\end{align*} (which is supplementary to the angle \begin{align*}\angle LOM\end{align*})
\begin{align*}180^\circ - 60^\circ = 120^\circ\end{align*}
C. \begin{align*}m \widehat{LMQ} = m \widehat{ML} + m \widehat{PM} + m \widehat{PQ}\!\\ = 60^\circ + 120^\circ + 60^\circ = 240^\circ\end{align*}
Example 2
Find \begin{align*}m \widehat{AB}\end{align*} in circle \begin{align*}O\end{align*}. The measures of all three arcs shown must add to \begin{align*}360^\circ\end{align*}.
All three arcs must add to \begin{align*}360^\circ\end{align*} because all three central angles add to \begin{align*}360^\circ\end{align*} (since they complete a circle.)
Fill in the arc measurements based on the picture above:
\begin{align*}m \angle AOB = m \widehat{AB} =\end{align*} _________________
\begin{align*}m \angle BOC = m \widehat{BC} =\end{align*} _________________
\begin{align*}m \angle AOC = m \widehat{AC} =\end{align*} _________________
We can add all three arcs together:
\begin{align*}m \widehat{AB}{\;\;} + m \widehat{BC}{\;\;} + m \widehat{AC}{\;\;\;} &= 360^\circ\\ x + 20 + 4x + 5 + 3x + 15 &= 360\\ 8x + 40 &= 360\\ 8x &= 320\\ x &= 40\end{align*}
We are looking for \begin{align*}m \widehat{AB}\end{align*} so we substitute \begin{align*}x = 40\end{align*} back into the arc measurement:
\begin{align*}m \widehat{AB} = x + 20 = 40 +20 = 60^\circ\end{align*}
Reading Check:
What other postulate is similar to the Arc Addition Postulate? Describe.
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Graphic Organizer for Lessons 1 – 2: PARTS OF A CIRCLE
Circle Part | Draw a Picture | What is it? | Is there a formula I need to know for this part? |
---|---|---|---|
Center | |||
Radius | |||
Diameter | |||
Circumference | |||
Area | |||
Central Angle | |||
Semicircle | |||
Minor Arc | |||
Major Arc | |||
Chord | |||
Secant |
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