Topics
13. Circles
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United States of America
Grades 9-12

13.03 Chords and angles

Lesson
Practice

Introduction

In the previous lesson, we began our exploration of types of lines and segments in circles with tangent lines. We will continue our exploration in this lesson with chords. As with tangents, we will learn about and apply several theorems related to chords and the segments and arcs they create.

Chords and angles

Between any two points on a circle, we can define the following line segment:

Chord

A line segment that connects two points on the arc of a circle. For example, \overline{AB} is a chord.

Points A and B on a circle. A segment is drawn from A to B.

Exploration

Using the applet below, move points C and D to change the lengths of the chords. Move point E to change the location of \overline{EF} around the circle. Move point B to change the size of the circle.

  1. What can you conclude about the arc lengths of the arcs EF and CD?
  2. How could we prove the arc lengths of arc EF and arc CD are congruent?
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The following theorems each relate to properties of one or more chords in a circle:

Congruent corresponding chords theorem

Two chords in a circle, or in congruent circles, are congruent if and only if the corresponding arcs are congruent

Circle C with chords A B and X Y. Minor arcs A B and X Y are congruent. Chords A B and X Y are congruent.
Corollary to the corresponding chords theorem

Two chords in a circle, or in congruent circles, are congruent if and only if the corresponding central angles are congruent

Circle C with chords A B and X Y. Chords A B and X Y are congruent. Central angles A C B and X C Y are congruent.
Equidistant chords theorem

In the same circle, or congruent circles, two chords are congruent if and only if they are equidistant from the center

Circle C with chords A B and X Y. Chords A B and X Y are congruent. The perpendicular distance from A B to C, and from X Y to C are congruent.

Examples

Example 1

Prove each of the following statements.

a

Prove that the corresponding central angles of two chords in a circle are congruent if the chords are congruent.

Worked Solution
Create a strategy

Using a particular diagram we are trying to prove that if \overline{AB} \cong \overline{CD}, then \angle BOA \cong \angle DOC.

Apply the idea

Given: \overline{AB} \cong \overline{CD}

To prove: \angle BOA \cong \angle DOC
StatementsReasons
1.\overline{AB} \cong \overline{CD}Given
2.\overline{OB} \cong \overline{OD}Both radii, radii are congruent
3.\overline{OA} \cong \overline{OC}Both radii, radii are congruent
4. \triangle BOA \cong \triangle DOCSide-Side-Side congruency theorem
5.\angle BOA \cong \angle DOCCorresponding parts of congruent triangles theorem
Reflect and check

The proof for if \angle BOA \cong \angle DOC, then \overline{AB} \cong \overline{CD} is very similar, but uses the Side-Angle-Side congruence test using the four radii and \angle BOA \cong \angle DOC.

It is important that we do not make assumptions about relationships based on a sketch. Although the central angles may appear to be vertical angles, we cannot use the congruent vertical angles theorem because the radii are not colinear. That would need to be included in the given information in order for us to use it.

b

Prove that the corresponding arcs of two chords in a circle are congruent if the chords are congruent.

Worked Solution
Create a strategy

Most of the proof comes from part (a), but we just add the calculation for arc length. We want to show: If \overline{AB} \cong \overline{CD}, then \overset{\large\frown}{AB} \cong \overset{\large\frown}{CD}.

Apply the idea

Using the corollary of the theorem, we have already proven:\text{If }\overline{AB} \cong \overline{CD}\text{, then }\angle BOA \cong \angle DOC.

From this, we know that:

\displaystyle \overset{\large\frown}{AB}\displaystyle =\displaystyle \dfrac{\theta}{360}2\pi rFormula for arc length
\displaystyle =\displaystyle \dfrac{m\angle BOA}{360}2\pi \cdot OASubstitution
\displaystyle \overset{\large\frown}{CD}\displaystyle =\displaystyle \dfrac{\theta}{360}2\pi rFormula for arc length
\displaystyle =\displaystyle \dfrac{m\angle DOC}{360}2\pi \cdot OCSubstitution

However, since \angle BOA \cong \angle DOC and \overline{OA} \cong \overline{OC}, we have that m\angle BOA = m \angle DOC and OA = OC, so

\displaystyle \overset{\large\frown}{AB}\displaystyle =\displaystyle \dfrac{m\angle DOC}{360}2\pi \cdot OCSubstitution
\displaystyle \overset{\large\frown}{AB}\displaystyle =\displaystyle \overset{\large\frown}{CD}Substitution

So if \overline{AB} \cong \overline{CD}, then \overset{\large\frown}{AB} \cong \overset{\large\frown}{CD} as required.

Reflect and check

To prove the other part of the biconditional statement, we just need to work backwards using the algebraic work.

Example 2

Given NP=4x and LM=6x-9, find LM.

Circle Q is drawn with chord P N and chord L M opposite each other. Segment Q S is drawn perpendicular to chord P N. Segment Q R is drawn perpendicular to chord L M. Segment Q S and segment Q R are marked congruent.
Worked Solution
Create a strategy

We have been given that the two chords are equidisant from the center of the circle. We can use the equidisant chords theorem to set up and equation so we can solve for x and then evaluate LM=6x-9.

Apply the idea
\displaystyle LM\displaystyle =\displaystyle NPEquidisant chords theorem and definition of congruence
\displaystyle 6x-9\displaystyle =\displaystyle 4xSubstitution
\displaystyle 2x\displaystyle =\displaystyle 9Add 9 and subtract 4x from both sides
\displaystyle x\displaystyle =\displaystyle \frac{9}{2}Divide both sides by 2
\displaystyle LM\displaystyle =\displaystyle 6\left(\dfrac{9}{2}\right)-9Substitute x into the equation for LM
\displaystyle =\displaystyle 27-9Evaluate the multiplication
\displaystyle =\displaystyle 18Evaluate the subtraction
Reflect and check

Since LM=NP, we can check the answer by substituting x=\dfrac{9}{2} into the equation for NP and seeing if it is equivalent to the length we found for LM.

\displaystyle NP\displaystyle =\displaystyle 4xEquation for NP
\displaystyle =\displaystyle 4\left(\dfrac{9}{2}\right)Substitute x=\dfrac{9}{2}
\displaystyle =\displaystyle 18Evaluate the multiplication

This shows LM=NP, so our answers are correct.

Example 3

In the diagram below, we are given that \angle BAC \cong \angle EAD, CB = 2x + 15, and ED = 5x -3.

Solve for x.

Two triangles, A B C and A E D, inside circle A. Segments C B and E D are chords of the circle. Angles E A D and B A C are congruent.
Worked Solution
Create a strategy

We know that \angle BAC \cong \angle EAD, and these are the central angles of the chords CB and DE. So by corollary to the corresponding chords theorem, we know that the chords are also congruent. We can now write an equation relating the two lengths of the chords and solve for x.

Apply the idea
\displaystyle 2x + 15\displaystyle =\displaystyle 5x-3Equating chord lengths
\displaystyle 2x+18\displaystyle =\displaystyle 5xAdd 3 to both sides
\displaystyle 18\displaystyle =\displaystyle 3xSubtract 2x from both sides
\displaystyle \dfrac{18}{3}\displaystyle =\displaystyle xDivide both sides by 3
\displaystyle x\displaystyle =\displaystyle 6Symmetric property of equality, evaluate the division
Idea summary

We can use the congruent corresponding chords theorem and its corollary to find angle measures, arc lengths, and chord lengths in circles. We can also use the equidistant chords theorem to find lengths of chords in circles.

Chords and arcs

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 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.
Perpendicular chord bisector theorem

A diameter bisects a chord and its corresponding arc if and only if the diameter is perpendicular to the chord

Circle C with diameter A B and chord X Y. X Y is perpendicular to A B. A B divides chord X Y into 2 congruent segments and minor arc X Y into 2 congruent arcs.

Examples

Example 4

In the diagram below, AC=k+5 and AB=3k+3.

a

Find BC.

Worked Solution
Create a strategy

We have been given that the chord \overline{AB} is perpendicular to a diameter of the circle, so this diameter bisects \overline{AB}. This means that \overline{AC} \cong \overline{BC} and 2AC=AB. We can use this to set up an equation and solve it.

Apply the idea
\displaystyle AB\displaystyle =\displaystyle 2ACPerpendicular chord bisector theorem
\displaystyle 3k+3\displaystyle =\displaystyle 2\left(k+5\right)Substitution
\displaystyle 3k+3\displaystyle =\displaystyle 2k+10Distribute the 2
\displaystyle k\displaystyle =\displaystyle 7Subtract 2k and 3 from both sides

Using that k=7, we can find that AC=7+5=12. Since we also know that \overline{AC} \cong \overline{BC}, \\BC=12.

b

Find m \overset{\large\frown}{AEB}.

Worked Solution
Create a strategy

By the perpendicular chord bisector theorem, we know m\overset{\large\frown}{BD}= m\overset{\large\frown}{AD}. We can use this to set up an equation and solve for x, then use the result to find m\overset{\large\frown}{AB}. This is the minor arc, but the question has asked us for the major arc m \overset{\large\frown}{AEB}. We can find this by subtracting m\overset{\large\frown}{AB} from 360\degree.

Apply the idea
\displaystyle 2x-8\displaystyle =\displaystyle 127-xPerpendicular chord bisector theorem
\displaystyle 3x\displaystyle =\displaystyle 135Add 8 and x to both sides
\displaystyle x\displaystyle =\displaystyle 45Divide both sides by 3

Now, we need to substitute this back into each expression to find m\overset{\large\frown}{AD} and m\overset{\large\frown}{DB}, and add them with the arc addition postulate to find m\overset{\large\frown}{AB}.

\displaystyle m\overset{\large\frown}{AB}\displaystyle =\displaystyle m\overset{\large\frown}{AD}+m\overset{\large\frown}{DB}Arc addition postulate
\displaystyle =\displaystyle \left(2x-8\right)+\left(127-x\right)Substitute expressions for arc measures
\displaystyle =\displaystyle \left(2\cdot 45-8\right)+\left(127-45\right)Substitute x=45
\displaystyle =\displaystyle 82+82Evalute parentheses
\displaystyle =\displaystyle 164\degreeEvaluate the addition

Lastly, we need to find m\overset{\large\frown}{AEB} by subtracting m\overset{\large\frown}{AB} from 360.

\displaystyle m\overset{\large\frown}{AEB}\displaystyle =\displaystyle 360-m\overset{\large\frown}{AB}
\displaystyle =\displaystyle 360-164
\displaystyle =\displaystyle 196\degree

Example 5

Consider the given diagram:

Circle Q with radii Q J, Q K, Q L and Q M placed clockwise on the circle. Angle J Q K has a measure of 115 degrees, K Q L has a measure of 9 degrees, and L Q M has a measure of 70 degrees.

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 JGK 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 6

Given \overline{PM} is a diameter, find m\overset{\large\frown}{NMP} in circle Z.

Worked Solution
Create a strategy

If we add \angle NZM and \angle MZL, we find the sum is \pi which means \overline{NL} is a diameter. Because we were given \overline{PM} is a diameter, we can confirm that \angle PZL and \angle NZM are vertical angles and are therefore, congruent.

Apply the idea

We know \angle PZL\cong \angle NZM by the vertical angles theorem, so m\angle PZL=\dfrac{\pi}{6}. By the congruent central angles theorem, we also know m\overset{\large\frown}{PL}\cong m\overset{\large\frown}{NM} since the measure of an arc is the same as the measure of its central angle. Now, we can use the arc addition postulate to find m\overset{\large\frown}{NMP}.

\displaystyle m\overset{\large\frown}{NMP}\displaystyle =\displaystyle m\overset{\large\frown}{NM}+m\overset{\large\frown}{ML}+m\overset{\large\frown}{LP}Arc addition postulate
\displaystyle =\displaystyle \dfrac{\pi}{6}+\dfrac{5\pi}{6}+\dfrac{\pi}{6}Substitute known values
\displaystyle =\displaystyle \dfrac{7\pi}{6}Evalute the addition
Reflect and check

Based on the diagram alone, we could not have assumed \angle PZL and \angle NZM were vertical angles. Diagrams are useful for helping visualize a direction for a proof or for solving a problem, but we cannot rely on the to tell us facts about measurements.

Idea summary

The measure of an arc is defined as the 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.

If a diameter is a perpendicular bisector of a chord, the perpendicular chord bisector theorem helps us find the lengths of the chords and arcs in the circle.

Outcomes

G.C.A.2

Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

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