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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.

13:45

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. Let’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 𝑂𝐢 is the perpendicular bisector of the chord 𝐴𝐡. 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 𝑂𝐴, we see that triangle 𝑂𝐢𝐴 is a right triangle. Using the Pythagorean theorem, 𝐴𝐢 squared plus 𝑂𝐢 squared is equal to 𝑂𝐴 squared. Since 𝐢 is the midpoint of the chord 𝐴𝐡, we also know that 𝐴𝐡 is equal to two multiplied by 𝐴𝐢. This means that if we are given the radius of the circle 𝑂𝐴 together with the distance of a chord from the center of the circle 𝑂𝐢, then we can calculate the length of the chord 𝐴𝐡.

Let’s now consider a second chord 𝐷𝐸 on the same diagram. Adding the perpendicular bisector of the chord 𝑂𝐹 together with the radius 𝑂𝐷, our diagram is as shown. Since 𝑂𝐴 and 𝑂𝐷 are radii of the circle, they have the same length. We wish to consider the relationship between the lengths of the chords 𝐴𝐡 and 𝐷𝐸. If we know that 𝐷𝐸 is further from the center than 𝐴𝐡, we can assume that 𝑂𝐢 is less than 𝑂𝐹. Comparing the half chords 𝐴𝐢 and 𝐷𝐹 and using the Pythagorean theorem once again, we have 𝐴𝐢 squared plus 𝑂𝐢 squared is equal to 𝑂𝐴 squared and 𝐷𝐹 squared plus 𝑂𝐹 squared is equal to 𝑂𝐷 squared.

We know that 𝑂𝐴 is equal to 𝑂𝐷. This means that the left-hand side of both equations must also be equal. 𝐴𝐢 squared plus 𝑂𝐢 squared is equal to 𝐷𝐹 squared plus 𝑂𝐹 squared. We can rearrange this equation such that 𝐴𝐢 squared minus 𝐷𝐹 squared is equal to 𝑂𝐹 squared minus 𝑂𝐢 squared. Since 𝑂𝐹 is greater than 𝑂𝐢, 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 𝐷𝐹 squared to both sides of this inequality, we have 𝐴𝐢 squared is greater than 𝐷𝐹 squared. And since 𝐴𝐢 and 𝐷𝐹 are both positive lengths, square rooting both sides gives us 𝐴𝐢 is greater than 𝐷𝐹.

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 𝐡𝐢 equals eight centimeters and 𝐡𝐴 equals seven centimeters. Which of the following is true? Is it (A) 𝐷𝑀 is equal to π‘‹π‘Œ, (B) 𝐷𝑀 is greater than π‘‹π‘Œ, or (C) 𝐷𝑀 is less than π‘‹π‘Œ?

Let’s begin by adding the lengths 𝐡𝐢 and 𝐡𝐴 to our diagram. These are the distances from the chords 𝐷𝑀 and π‘‹π‘Œ, respectively, to the center of the circle 𝐡. 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‘Œ is seven centimeters from the center. This is the length of 𝐡𝐴. The chord 𝐷𝑀, on the other hand, is eight centimeters from the center. This means that π‘‹π‘Œ is closer to the center of the circle than 𝐷𝑀. As this chord will have a greater length, we can conclude that π‘‹π‘Œ is greater than 𝐷𝑀. From the three options listed, the correct answer is option (C) 𝐷𝑀 is less than π‘‹π‘Œ.

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

In the diagram drawn, we assume that the chords 𝐴𝐡 and 𝐷𝐸 are equidistant from the center. This means that 𝑂𝐢 is equal to 𝑂𝐹. The two radii 𝑂𝐴 and 𝑂𝐷 will also be equal in length. This means that two sides of our right triangles 𝑂𝐢𝐴 and 𝑂𝐹𝐷 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 𝐴𝐢 is equal to the half chord 𝐷𝐹. This in turn leads to the fact that the chords 𝐴𝐡 and 𝐷𝐸 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. Let’s now consider an example where we need to find the missing length of a chord in a given diagram.

Given that 𝑀𝐢 is equal to 𝑀𝐹, which is equal to three centimeters, 𝐴𝐢 is equal to four centimeters, line segment 𝑀𝐢 is perpendicular to line segment 𝐴𝐡, and line segment 𝑀𝐹 is perpendicular to line segment 𝐷𝐸, find the length of the line segment 𝐷𝐸.

We are told in the question that 𝑀𝐢 is equal to 𝑀𝐹. And this means that the two chords 𝐴𝐡 and 𝐷𝐸 are equidistant from the center 𝑀. 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 𝐴𝐡 is equal to the length 𝐷𝐸. We also know that 𝑀𝐢 is the perpendicular bisector of 𝐴𝐡. And this means that 𝐴𝐡 is equal to two multiplied by 𝐴𝐢. Since 𝐴𝐢 is equal to four centimeters, 𝐴𝐡 is equal to eight centimeters. We can therefore conclude that the length of the chord 𝐷𝐸 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 𝐴𝐡 and 𝐢𝐷 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 𝑂𝐸 and 𝑃𝐹, respectively, Adding the radii 𝑂𝐴 and 𝑃𝐢, 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 𝐴𝐸 is equal to 𝐢𝐹. This is because 𝐴𝐸 is equal to a half of 𝐴𝐡 and 𝐢𝐹 is equal to a half of 𝐢𝐷.

Using our knowledge of the Pythagorean theorem, the third sides of our right triangles must also be equal in length. The length 𝑂𝐸 is equal to the length 𝑃𝐹. 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 𝐴𝐡 is equal to 𝐢𝐷, 𝑀𝐢 is equal to 10 centimeters, and 𝐷𝐹 is equal to eight centimeters, find the length of line segment 𝑀𝐸.

In this question, we are trying to calculate the length of 𝑀𝐸, which is the distance from the chord 𝐴𝐡 to the center of the circle 𝑀. 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 𝐴𝐡 and 𝐢𝐷 are equal in length. This means that the length 𝑀𝐹 must be equal to 𝑀𝐸. The line segment 𝑀𝐸 perpendicularly bisects the chord 𝐴𝐡. Likewise, 𝑀𝐹 is the perpendicular bisector of 𝐢𝐷. Since we are told 𝐷𝐹 is equal to eight centimeters, 𝐢𝐹, 𝐴𝐸, and 𝐡𝐸 are all also equal to eight centimeters.

Our next step is to consider the right triangle 𝑀𝐹𝐢. Using the Pythagorean theorem, 𝑀𝐹 squared plus 𝐢𝐹 squared is equal to 𝑀𝐢 squared. By subtracting 𝐢𝐹 squared from both sides and substituting in the values of 𝐢𝐹 and 𝑀𝐢, we have 𝑀𝐹 squared is equal to 10 squared minus eight squared. This is equal to 36. Square rooting both sides of this equation and knowing that 𝑀𝐹 must be a positive answer, we have 𝑀𝐹 is equal to six. Since 𝑀𝐹 is equal to six centimeters, 𝑀𝐸 must also be equal to six centimeters. The perpendicular distance from the center of the chord 𝐴𝐡 to the center of the circle 𝑀, which is the line segment 𝑀𝐸, 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.

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