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12. Circles
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Geometry

12.02 Measures of arcs

Lesson
Practice

Arcs of central angles

Central angle

An angle that has its vertex at the center of a circle with radii as its sides.

A circle with two radii drawn. The angle between them is marked

Arcs of a circle can be further classified as follows:

Semicircle

An arc of a circle with endpoints on the diameter of the circle.

An arc of a circle whose endpoints lie on a diameter.

Minor arc

An arc of a circle smaller than a semicircle.

An arc smaller than a semicircle.

Major arc

An arc of a circle larger than a semicircle.

An arc larger than a semicircle.

The notation we use to denote a minor arc with endpoints at A and B is \overset{\large\frown}{AB}.

To distinguish between a major arc and a minor arc, we use a third point that lies between the endpoints. If the endpoints of an arc are A and B and point P lies between them on the major arc, we use the notation \overset{\large\frown}{APB}.

Measure of an arc

The measure of an arc is equal to the measure of its central angle

The measure of an arc is different from the length of the arc. While arc length refers to the distance from one endpoint of the arc to the next, the measure of an arc refers to the measure of its central angle. We always use the notation m\overset{\large\frown}{AB} when talking about arc measure and \overset{\large\frown}{AB} when talking about arc length.

Adjacent arc measures can be combined by the following postulate:

Arc addition postulate

The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs

Circle C with minor arcs A B and B D.

This theorem is helpful to connect the central angle with the minor arc.

Congruent central angles theorem

In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding central angles are congruent

Circle C with diameters A Y and X B. Central angles A C B and X C Y are congruent. Minor arcs A B and X Y are congruent.

Examples

Example 1

Consider the given diagram:

Circle Q. 4 Radii are drawn from points marked from the left counter clockwise: J, K, L and M.The angle formed by JQK is 115 degrees, the angled formed by KQL is 9 degrees.
a

Find \angle JQM given m\overset{\large\frown}{JM}=166 \degree

Worked Solution
Create a strategy

The measure of \angle JQM is equivalent to m\overset{\large\frown}{JM}.

Apply the idea

Since \angle JQM is a central angle, we know that the measure of corresponding arc, m\overset{\large\frown}{JM}, are equivalent. So \angle JQM = 166\degree.

b

Find m\overset{\large\frown}{JL}

Worked Solution
Create a strategy

The measure of arc JL is made up of arc JK and arc KL, so we will need to use the arc addition postulate to add the arc measures. The measure of arc JK is the measure of \angle JQK and the measure of arc KL is the measure of \angle KQL.

Apply the idea
\displaystyle m\overset{\large\frown}{JL}\displaystyle =\displaystyle m\overset{\large\frown}{JK} + m\overset{\large\frown} {KL}Arc addition postulate
\displaystyle m\overset{\large\frown}{JL}\displaystyle =\displaystyle 115 \degree + 9 \degreeSubstitute known values
\displaystyle =\displaystyle 124 \degreeEvaluate the addition
Reflect and check

The corresponding major arc of arc JL is arc JML. A minor arc and its corresponding major arc add to 360\degree, so we know that m\overset{\large\frown}{JML}=360-124=236\degree.

Example 2

Let m\angle EDH=\left(6x-5\right)\degree and m\overset{\large\frown}{FG} = \left(5x+20\right)\degree.

Circle D. 4 radii drawn from points E,F,G, and H marked on the circle clockwise starting from the left. Arc EH is congruent to Arc FG.
a

Solve for x.

Worked Solution
Create a strategy

Since \angle EDH is a central angle, we know its measure is equivalent to the measure of its corresponding arc, \overset{\large\frown}{EH}. We can also see from image that \overset{\large\frown}{EH} is congruent to \overset{\large\frown}{FG}.

Apply the idea

First, set up the equation, knowing m\angle EDH will be equal to m\overset{\large\frown}{FG}. Then solve for x.

\displaystyle 6x-5\displaystyle =\displaystyle 5x+20Set up equation
\displaystyle x-5\displaystyle =\displaystyle 20Subtract 5x from both sides
\displaystyle x\displaystyle =\displaystyle 25Add 5 to both sides
Reflect and check

We can confirm our solution by substituting x=25 back into the original equation.

\displaystyle 6x-5\displaystyle =\displaystyle 5x+20Write original equation
\displaystyle 6\left(25\right)-5\displaystyle =\displaystyle 5\left(25\right)+20Substitute x=25
\displaystyle 150-5\displaystyle =\displaystyle 125+20Evaluate the multiplication
\displaystyle 145\displaystyle =\displaystyle 145Simplify

Substituting our x-value has resulted in a true statement, we have confirmed our solution.

b

Find m\angle EDH

Worked Solution
Create a strategy

Since we have an expression representing m\angle EDH, we can substitute the value of x we found in part (a) to find the measure of the central angle.

Apply the idea
\displaystyle m\angle EDH\displaystyle =\displaystyle 6x-5Given
\displaystyle {}\displaystyle =\displaystyle 6\left(25\right)-5Substitute x=25
\displaystyle {}\displaystyle =\displaystyle 150-5Evaluate the multiplication
\displaystyle {}\displaystyle =\displaystyle 145Evaluate the subtraction

The m\angle EDH=145\degree

Reflect and check

Since the measure of the central angle m\angle EDH=145\degree, this means that its corresponding arc m\overset{\large\frown}{EH}=145\degree. From the image we can also tell that the central angle m\angle FDG=145\degree and its corresponding arc m\overset{\large\frown}{FG}=145\degree.

Example 3

The area of a sector of a circle with radius 8 \text{ cm} is \dfrac{16}{3}\pi \text{ cm}^2

a

Find the measure of the central angle

Worked Solution
Create a strategy

We know the formula for the arc length of a sector to be A=\dfrac{\theta}{360}\cdot \pi r^2. We can substitute the known values of sector area and radius to solve for the central angle that we will call \theta.

Apply the idea
\displaystyle A\displaystyle =\displaystyle \dfrac{\theta}{360}\cdot \pi r^2Formula for sector area
\displaystyle \dfrac{16\pi}{3}\displaystyle =\displaystyle \dfrac{\theta}{360}\cdot \pi\cdot8^2Substitute r=8 \text{ cm} and A=\dfrac{16\pi}{3}
\displaystyle \dfrac{16\pi}{3}\displaystyle =\displaystyle \dfrac{\theta}{360}\cdot 64\piEvaluate the exponent
\displaystyle \dfrac{16\pi}{3\cdot64\pi}\displaystyle =\displaystyle \dfrac{\theta}{360}Divide both sides by 64\pi
\displaystyle \dfrac{16 \cancel{\pi}}{3\cdot64\cancel{\pi}}\displaystyle =\displaystyle \dfrac{\theta}{360}Divide out common factor of \pi
\displaystyle \dfrac{1}{12}\displaystyle =\displaystyle \dfrac{\theta}{360}Simplify the fraction
\displaystyle 30\displaystyle =\displaystyle \thetaMultiply both sides by 360

We have just found that the central angle, \theta = 30\degree.

b

Find the measure of the corresponding arc

Worked Solution
Create a strategy

The measure of the corresponding arc is equal to the measure of its central angle.

Apply the idea

We found in part (a) that the measure of the central angle is 30\degree, which means its corresponding arc also measures 30\degree.

Idea summary

\text{Measure of arc }=\text{ Measure of its central angle}

By the congruent central angles theorem, two minor arcs are congruent if and only if their corresponding central angles are congruent.

We can find the sum of adjacent arcs using the arc addition postulate. By this, we know the minor arc of a central angle and its corresponding major arc will sum to 360\degree or 2\pi radians.

Arcs of inscribed angles

Angles formed by chords of a circle are known as inscribed angles.

Inscribed angle in a circle

An angle whose vertex is a point on the circle and whose sides contain chords of the circle.

Two chords are on a circle which both have an endpoint at the same point on the circle. The angle between the chords is an inscribed angle in the circle.

Exploration

Drag the slider to move point B around the circle to set the angle measures.

Drag point A to rotate the central angle and C to rotate the inscribed angle.

Loading interactive...
  1. What relationship do you notice between the central angle and the inscribed angle?
  2. How does this relationship translate to the inscribed angle and the arc it intercepts?

The following theorems relate to angles inscribed in circles:

Inscribed angle theorem

If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc

An angle with a measure of theta degrees inscribed in a circle. The angle intercepts the arc A B with a measure of 2 theta degrees.
Congruent inscribed angle theorem

If two inscribed angles of a circle intercept the same arc, then the angles are congruent

A circle with points A, C, D, and B placed clockwise on a circle. Chords A C, A D, C B, and B D. Chords A D and C B intersect at a point. Angles A C B and A D B are congruent.

Examples

Example 4

Solve for x.

Angle A T B with a measure of negative 10 plus x degrees inscribed in a circle. Angle A T B intercepts an arc A B which has a measure of 4 plus x degrees.
Worked Solution
Create a strategy

We can use the inscribed angle theorem to write an equation and then solve for x.

Apply the idea
\displaystyle 2(m\angle{ATB}) \displaystyle =\displaystyle m \overset{\large\frown}{AB}
\displaystyle 2\left(-10+x\right)\displaystyle =\displaystyle 4+xSubstitution
\displaystyle -20+2x\displaystyle =\displaystyle 4+xDistribute the 2
\displaystyle 2x\displaystyle =\displaystyle 24+xAdd 20 to both sides
\displaystyle x\displaystyle =\displaystyle 24Subtract x from both sides

x=24

Reflect and check

The value of x is 24, which means the inscribed angle m \angle ATB = 14 \degree and the intercepted arc m\overset{\large\frown}{AB} = 28 \degree.

Example 5

Given m\angle CEB = 4x + 11 and m\angle CDB = 12x - 5. Find m\angle CDB.

Circle A with points C, E, D, and B placed clockwise on the circle. Chords C E, C D, C B, E B, and D B are drawn.
Worked Solution
Create a strategy

By the congruent inscribed angle theorem, we know m\angle CEB = m\angle CDB. We want to write an equation relating the two angles and then solve for x.

Apply the idea
\displaystyle 4x + 11\displaystyle =\displaystyle 12x - 5Congruent inscribed angle theorem
\displaystyle 4x + 16\displaystyle =\displaystyle 12xAdd 5 to both sides
\displaystyle 16\displaystyle =\displaystyle 8xSubtract 4x from both sides
\displaystyle 2\displaystyle =\displaystyle xDivide both sides by 8
\displaystyle x\displaystyle =\displaystyle 2Symmetric property of equality

We have established x = 2, so we can substitute that into the equation for \angle CDB.

Substituting x = 2 into 12x - 5 we get 12\left(2\right) - 5 = 19. Therefore, m\angle CDB = 19 \degree.

Reflect and check

Note that since m\angle CDB = m\angle CEB, we could have used the expression for m\angle CEB to calculate the size of the angle instead:

\displaystyle m\angle CEB\displaystyle =\displaystyle 4x + 11Given
\displaystyle =\displaystyle 4\left(2\right) + 11Substitute x = 2
\displaystyle =\displaystyle 19Simplify the expression

which is the same result.

Idea summary

\text{Measure of inscribed angle }= \dfrac{1}{2} \cdot \text{Measure of the intercepted arc}

If two inscribed angles intercept the same arc, the angles are congruent.

Outcomes

G.PC.3

The student will solve problems, including those in context, by applying properties of circles.

G.PC.3b

Solve for arc measures and angles in a circle formed by central angles.

G.PC.3c

Solve for arc measures and angles in a circle involving inscribed angles.

G.PC.3f

Apply arc length or sector area to solve for an unknown measurement of the circle including the radius, diameter, arc measure, central angle, arc length, or sector area.

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