In this explainer, we will learn how to identify the relationship between chords that are equal or different in length and the center of a circle and use the properties of the chords in congruent circles to solve problems.
We begin by recalling that perpendicular bisectors of chords go through the center of the circle. Let us draw a diagram portraying this fact.
In the diagram above, the blue line segment perpendicularly bisects chord . We note that this line goes through the center and, hence, defines the perpendicular distance between the center and the chord.
Definition: Distance of a Chord from the Center
The distance of a chord from the center of the circle is measured by the length of the line segment from the center that intersects perpendicularly with the chord.
From the diagram above, let us label the midpoint of chord , which is where the blue line perpendicularly intersect with the chord. Also, we will add radius .
Since is a right triangle, we can use the Pythagorean theorem to find length from radius and distance . Since is the midpoint of chord , we know that . Hence, if we are given the radius of the circle and the distance of a chord from the center of the circle, we can use this method to find the length of the chord. Rather than explicitly writing out this computation, we will focus on the qualitative relationship between the lengths of chords and their distance from the center of the circle in this explainer.
Consider two different chords in the same circle as in the diagram below.
Since and are radii of the same circle, they have the same length. We want to know the relationship between the lengths of chords and if we know that is farther from the center than . In other words, we assume . Instead of comparing the full lengths of the two chords, we can compare the half-chords and . Using the Pythagorean theorem, we can write
We know that , so the left-hand sides of both equations must be equal to each other:
This equation can be rearranged to say
Our assumption that leads to , so the left-hand side of this equation must be positive. This means
Since and are positive lengths, we can take the square root of both sides of the inequality to obtain . This leads to the following statement.
Theorem: Relationship between the Lengths of Chords and Their Distance from the Center
Consider two chords in the same circle whose distances from the center are different. The chord that is closer to the center of the circle has a greater length than the other.
This theorem allows us to compare the lengths of chords in the same circle based on their distance from the center of the circle. In our first example, we will apply this theorem to obtain an inequality involving lengths.
Example 1: Comparing Chord Lengthes based on their Distances from the Center
Supposed that and . Which of the following is true?
Answer
We recall that for two chords in the same circle, the chord that is closer to the center of the circle has a greater length than the other. We also know that the distance of a chord from the center of the circle is measured by the length of the line segment from the center intersecting perpendicularly with the chord.
In this example, we have two chords and . Since intersects perpendicularly with chord , length is the distance of this chord from the center. Similarly, length is the distance of chord from the center. Based on the given information, we note that , which means that chord is closer to the center. Hence, the length of chord is greater than that of the other chord.
The true option is C, which states that .
In the next example, we will find the range of an unknown variable defining lengths using the relationship between chords and the center of the circle.
Example 2: Finding the Range of Values of an Unknown That Satisfy Given Conditions
If , find the range of values of that satisfy the data represented.
Answer
We recall that for two chords in the same circle, the chord that is closer to the center of the circle has a greater length than the other. We also know that the distance of a chord from the center of the circle is measured by the length of the line segment from the center intersecting perpendicularly with the chord.
In this example, we have two chords, and . Since intersects perpendicularly with chord , length is the distance of this chord from the center. Similarly, length is the distance of chord from the center. Since we are given , we know that chord is closer to the center. This leads to the fact that chord has a greater length than chord .
In the given diagram, we note that and . Hence, the inequality can be written as
However, this only provides the lower bound for . To identify the upper bound for , we should ask what the maximum length of chord is. Since the length of a chord is larger when it is closer to the center, the longest chord should occur when the distance from the center is zero. If the distance of a chord from the center is zero, the chord should contain the center. In this case, the chord is a diameter of the circle. Since the radius of the circle is 33 cm, its diameter is . This tells us that the length of cannot exceed 66 cm. Additionally, since in the given diagram does not contain the center , we know that the length of chord must be strictly less than 66 cm. Hence,
This gives us the upper bound for . Combining both lower and upper bounds, we have
In interval notation, this is written as .
In previous examples, we considered the relationship between the lengths of two chords in the same circle and their distances from the center of the circle when the distances are not the equal. Recall that two circles are congruent to each other if the measures of their radii are equal. Since the proof of this relationship only uses the fact that the radii of the circle have equal lengths, this relationship can extend to two chords from two congruent circles.
What can we say about the lengths of chords in the same circle, or in congruent circles, if their distances from the respective centers are equal? It is not difficult to modify the previous discussion to fit this particular case. Consider the following diagram.
We assume that chords and are equidistant from the center, which means . We also know that the radii are of the same length, thus . This tells us that the hypotenuse and one other side of the two right triangles and are equal. Since the lengths of the remaining sides can be obtained using the Pythagorean theorem, the lengths of the third sides, and , must also be equal. Since these lengths are half of those of the chords, the two chords must have equal lengths. This result can be summarized as follows.
Theorem: Equidistant Chords in Congruent Circles
Consider two chords in the same circle, or in congruent circles. If they are equidistant from the center of the circle, or from the respective centers of the circles, then their lengths are equal.
In the next example, we will use this relationship to find a missing length of a chord in a given diagram.
Example 3: Finding a Missing Length Using Equidistant Chords from the Center of a Circle
Given that , , , and , find the length of .
Answer
We recall that two chords in the same circle that are equidistant from the center of the circle have equal lengths. We also know that the distance of a chord from the center of the circle is measured by the length of the line segment from the center intersecting perpendicularly with the chord.
In this example, we have two chords, and . Since intersects perpendicularly with chord , length is the distance of this chord from the center. Similarly, length is the distance of chord from the center. From the given information, we note that , so the two chords are equidistant from the center of the circle. Hence, the two chords must have equal lengths, .
In the diagram above, we are given that . We recall that the perpendicular bisector of a chord passes through the center of the circle. Since is perpendicular to chord and passes through center of the circle, it must be the perpendicular bisector of chord . In particular, this means that is the midpoint of , which gives us . Since , we also know that . Hence,
This tells us that the length of is 8 cm. Since we know , we conclude that the length of is 8 cm.
So far, we have discussed implications for the lengths of chords depending on their distance from the center of the circle. We now turn our attention to the converse relationship. More specifically, if we know that two chords in two congruent circles have equal lengths, what can we say about the distance of the chords from the respective centers of the circles? Let us consider the following diagram.
We can label the midpoints of both chords, which are where the blue lines intersect with the chords perpendicularly. Also, we add radii and to the diagram. Since the circles are congruent, we know that the lengths of the radii are equal, which leads to as seen in the diagram below.
We know that and are midpoints of the chords so
Since we are assuming that the chords have equal lengths, we know that as marked in the diagram above. This tells us that the hypotenuse and one other side of the two right triangles and are equal. Since the lengths of the remaining sides can be obtained using the Pythagorean theorem, the lengths of the third sides must also be equal. This tells us
In other words, the distances of the chords from the respective centers are equal. We can summarize this result as follows.
Theorem: Chords of Equal Lengths in Congruent Circles
Two chords of equal lengths in the same circle, or in congruent circles, are equidistant from the center of the circle, or the respective centers of the circles.
Let us consider an example where we need to use this statement together with other properties of the chords of a circle to find a missing length.
Example 4: Finding a Missing Length Using Equal Chords
Given that , , and , find the length of .
Answer
We recall that two chords of equal lengths in the same circle are equidistant from the center of the circle. We also know that the distance of a chord from the center of the circle is measured by the length of the line segment from the center intersecting perpendicularly with the chord.
In this example, we have two chords, and . Since intersects perpendicularly with chord , length is the distance of this chord from the center. Similarly, the length is the distance of chord from the center. Since we are given , we know that the chords have equal lengths. This leads to the fact that the chords are equidistant from the center:
Since we are looking for length , it suffices to find length instead. We note that is a side of the right triangle , whose hypotenuse is given by . If we can find the length of side , then we can apply the Pythagorean theorem to find the length of the third side, .
To find length , we recall that the perpendicular bisector of a chord goes through the center of the circle. Since perpendicularly intersects chord and goes through center , it is the perpendicular bisector of the chord. Hence, . Since , we obtain .
Applying the Pythagorean theorem to ,
Substituting and into this equation,
Since is a positive length, we can take the square root to obtain
Remember that since , we conclude that the length of is 6 cm.
In our final example, we will use the relationship between lengths of chords and their distances from the center of the circle to identify a missing angle.
Example 5: Finding the Measure of an Angle in a Triangle inside a Circle Where Two of Its Vertices Intersect with Chords and Its Third Is the Circleβs Center
Find .
Answer
We recall that two chords of equal lengths in the same circle are equidistant from the center of the circle. We also know that the distance of a chord from the center of the circle is measured by the length of the line segment from the center intersecting perpendicularly with the chord.
In this example, we have two chords and that have equal lengths. We recall that the perpendicular bisector of a chord goes through the center of the circle. Since and are midpoints of the two chords and is the center of the circle, line segments and must be the perpendicular bisectors of the two chords. In particular, these lines intersect perpendicularly with the respective chords. This tells us that and are the respective distances of chords and from the center of the circle.
Since the two chords have equal lengths, they must be equidistant from the center. This tells us
This also tells us that two sides of triangle have equal lengths. In other words, is an isosceles triangle. Hence,
We also know that the sum of the interior angles of a triangle is equal to . We can write
We know that and also . Substituting these expressions into the equation above,
Therefore, .
Let us finish by recapping a few important concepts from this explainer.
Key Points
- The distance of a chord from the center of the circle is measured by the length of the line segment from the center intersecting perpendicularly with the chord.
- Consider two chords in the same circle, or in two congruent circles, whose distances from the center, or the respective centers, are different. The chord that is closer to the respective center is of greater length than the other.
- Consider two chords in the same circle, or in congruent circles. If they are equidistant from the center of the circle, or from the respective centers of the circles, their lengths are equal.
- Two chords of equal lengths in the same circle, or in congruent circles, are equidistant from the center of the circle, or the respective centers of the circles.
