# Lesson Video: Relationships between Chords and the Center of a Circle Mathematics

In this video, 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.

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### Video Transcript

In this video, we will learn how to identify the relationship between chords that are equal or different in length and the center of the circle and use the properties of chords in congruent circles to solve problems. We begin by recalling that perpendicular bisectors of chords go through the center of the circle as shown. Letβs now consider this in more detail and how it leads to definitions and theorems we will use in this video.

In the diagram shown, the line segment ππΆ is the perpendicular bisector of the chord π΄π΅. This leads us to the following definition. 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. Adding the radius ππ΄, we see that triangle ππΆπ΄ is a right triangle. Using the Pythagorean theorem, π΄πΆ squared plus ππΆ squared is equal to ππ΄ squared. Since πΆ is the midpoint of the chord π΄π΅, we also know that π΄π΅ is equal to two multiplied by π΄πΆ. This means that if we are given the radius of the circle ππ΄ together with the distance of a chord from the center of the circle ππΆ, then we can calculate the length of the chord π΄π΅.

Letβs now consider a second chord π·πΈ on the same diagram. Adding the perpendicular bisector of the chord ππΉ together with the radius ππ·, our diagram is as shown. Since ππ΄ and ππ· are radii of the circle, they have the same length. We wish to consider the relationship between the lengths of the chords π΄π΅ and π·πΈ. If we know that π·πΈ is further from the center than π΄π΅, we can assume that ππΆ is less than ππΉ. Comparing the half chords π΄πΆ and π·πΉ and using the Pythagorean theorem once again, we have π΄πΆ squared plus ππΆ squared is equal to ππ΄ squared and π·πΉ squared plus ππΉ squared is equal to ππ· squared.

We know that ππ΄ is equal to ππ·. This means that the left-hand side of both equations must also be equal. π΄πΆ squared plus ππΆ squared is equal to π·πΉ squared plus ππΉ squared. We can rearrange this equation such that π΄πΆ squared minus π·πΉ squared is equal to ππΉ squared minus ππΆ squared. Since ππΉ is greater than ππΆ, the right-hand side of our equation must be greater than zero. This means that the left-hand side must also be greater than zero or positive. Adding π·πΉ squared to both sides of this inequality, we have π΄πΆ squared is greater than π·πΉ squared. And since π΄πΆ and π·πΉ are both positive lengths, square rooting both sides gives us π΄πΆ is greater than π·πΉ.

This leads us to the following theorem. If we consider two chords in the same circle whose distances from the center are different, then the chord which is closer to the center of the circle has a greater length than the other. We will now apply this theorem to a specific example.

Suppose that π΅πΆ equals eight centimeters and π΅π΄ equals seven centimeters. Which of the following is true? Is it (A) π·π is equal to ππ, (B) π·π is greater than ππ, or (C) π·π is less than ππ?

Letβs begin by adding the lengths π΅πΆ and π΅π΄ to our diagram. These are the distances from the chords π·π and ππ, respectively, to the center of the circle π΅. We recall that the chord that is closer to the center of the circle has a greater length. From the diagram, we see that the chord ππ is seven centimeters from the center. This is the length of π΅π΄. The chord π·π, on the other hand, is eight centimeters from the center. This means that ππ is closer to the center of the circle than π·π. As this chord will have a greater length, we can conclude that ππ is greater than π·π. From the three options listed, the correct answer is option (C) π·π is less than ππ.

So far, we have only considered the situation where the distances of two chords from the center of a circle are not equal. Letβs now consider what happens when the two chords are equidistant from the center.

In the diagram drawn, we assume that the chords π΄π΅ and π·πΈ are equidistant from the center. This means that ππΆ is equal to ππΉ. The two radii ππ΄ and ππ· will also be equal in length. This means that two sides of our right triangles ππΆπ΄ and ππΉπ· are equal. Using our knowledge of the Pythagorean theorem, this means that the length of the third sides of our triangle must also be equal. The half chord π΄πΆ is equal to the half chord π·πΉ. This in turn leads to the fact that the chords π΄π΅ and π·πΈ are equal in length. This can be summarized as follows. If we consider two chords in the same circle and if they are equidistant from the center of the circle, then their lengths are equal.

Whilst we will not see an example in this video, it is also important to note that this theorem holds for congruent circles. In that case, if the chords are equidistant from the respective centers of the circles, then the lengths are equal. Letβs now consider an example where we need to find the missing length of a chord in a given diagram.

Given that ππΆ is equal to ππΉ, which is equal to three centimeters, π΄πΆ is equal to four centimeters, line segment ππΆ is perpendicular to line segment π΄π΅, and line segment ππΉ is perpendicular to line segment π·πΈ, find the length of the line segment π·πΈ.

We are told in the question that ππΆ is equal to ππΉ. And this means that the two chords π΄π΅ and π·πΈ are equidistant from the center π. They are both three centimeters away from the center. We recall the theorem that states that if two chords are equidistant from the center, they are also equal in length. This means that in this question, the length π΄π΅ is equal to the length π·πΈ. We also know that ππΆ is the perpendicular bisector of π΄π΅. And this means that π΄π΅ is equal to two multiplied by π΄πΆ. Since π΄πΆ is equal to four centimeters, π΄π΅ is equal to eight centimeters. We can therefore conclude that the length of the chord π·πΈ is eight centimeters.

So far in this video, we have discussed the lengths of chords depending on their distance from the center of the circle. We will now consider the converse relationship. In the diagram shown, we have two congruent circles. We will consider the case where the chords π΄π΅ and πΆπ· are equal in length and, more importantly, what this says about the distance of the chords from their respective centers. In this example, these are the lengths ππΈ and ππΉ, respectively, Adding the radii ππ΄ and ππΆ, we know that these lengths must be equal, as the circles are congruent. As the chords are equal in length, the half chords must also be equal in length such that π΄πΈ is equal to πΆπΉ. This is because π΄πΈ is equal to a half of π΄π΅ and πΆπΉ is equal to a half of πΆπ·.

Using our knowledge of the Pythagorean theorem, the third sides of our right triangles must also be equal in length. The length ππΈ is equal to the length ππΉ. In other words, the distance of the chords from their respective centers are equal. This can be summarized as follows. Two chords of equal lengths in the same circle or congruent circles are equidistant from the center of the circle or the respective centers.

In our final example, we will use this statement to find a missing length.

Given that π΄π΅ is equal to πΆπ·, ππΆ is equal to 10 centimeters, and π·πΉ is equal to eight centimeters, find the length of line segment ππΈ.

In this question, we are trying to calculate the length of ππΈ, which is the distance from the chord π΄π΅ to the center of the circle π. We begin by recalling that two chords of equal lengths are equidistant from the center. And in this question, we are told that the two chords π΄π΅ and πΆπ· are equal in length. This means that the length ππΉ must be equal to ππΈ. The line segment ππΈ perpendicularly bisects the chord π΄π΅. Likewise, ππΉ is the perpendicular bisector of πΆπ·. Since we are told π·πΉ is equal to eight centimeters, πΆπΉ, π΄πΈ, and π΅πΈ are all also equal to eight centimeters.

Our next step is to consider the right triangle ππΉπΆ. Using the Pythagorean theorem, ππΉ squared plus πΆπΉ squared is equal to ππΆ squared. By subtracting πΆπΉ squared from both sides and substituting in the values of πΆπΉ and ππΆ, we have ππΉ squared is equal to 10 squared minus eight squared. This is equal to 36. Square rooting both sides of this equation and knowing that ππΉ must be a positive answer, we have ππΉ is equal to six. Since ππΉ is equal to six centimeters, ππΈ must also be equal to six centimeters. The perpendicular distance from the center of the chord π΄π΅ to the center of the circle π, which is the line segment ππΈ, is equal to six centimeters.

We will now finish this video by summarizing the key points from it. 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. When two chords whose distances from the center of a circle are different, then the chord which is closer to the center has a greater length. If two chords are equidistant from the center of a circle, their lengths are equal. The converse of this is also true. Two chords of equal lengths in the same circle are equidistant from the center. It is important to note that the last three statements are also true for two congruent circles.