Lesson Video: Perpendicular Bisector of a Chord | Nagwa Lesson Video: Perpendicular Bisector of a Chord | Nagwa

# Lesson Video: Perpendicular Bisector of a Chord Mathematics

In this video, we will learn how to use the theory of the perpendicular bisector of a chord from the center of a circle and its converse to solve problems.

14:11

### Video Transcript

In this video, we will learn how to use the theory of the perpendicular bisector of a chord from the center of a circle and its converse to solve problems. Letβs begin by recalling how we can define a radius, a chord, and a diameter. A radius is a line segment which has one end at the center of the circle and the other on the circumference. We define a chord as any straight line segment whose endpoints both lie on the circumference of the same circle. The diameter is a special type of chord which passes through the center of the circle. We can also consider this to be made up of two radii.

We will now consider what a perpendicular bisector of a chord looks like. In the diagram drawn, we have the chord π΅πΆ together with its perpendicular bisector. We will look at three theorems in this video. And in each case, we need to consider the center of the circle π΄ together with the radii π΄π΅ and π΄πΆ.

Our first theorem states that if we have a circle with center π΄ containing a chord π΅πΆ, then the straight line that passes through π΄ and bisects the chord π΅πΆ is perpendicular to π΅πΆ. The second theorem is very similar. If we have a circle with center π΄ containing a chord π΅πΆ, then the straight line that passes through π΄ and is perpendicular to π΅πΆ also bisects π΅πΆ. Our third theorem is the converse to the chord bisector theorem. This states that if we have a circle with center π΄ containing a chord π΅πΆ, then the perpendicular bisector of π΅πΆ passes through π΄.

It is important to note that the perpendicular bisector of a chord creates two congruent triangles. In the diagram drawn, these are triangles π΄π·πΆ and π΄π·π΅. We will now look at some examples where we can use the theorems discussed to find missing lengths and angles.

Given π΄π equals 200 centimeters and ππΆ equals 120 centimeters, find the length of the line segment π΄π΅.

In the diagram shown, we have a circle with center π. The chord π΄π΅ is bisected by the line segment ππ· at the point πΆ. Applying the chord bisector theorem, which states that if we have a circle with center π containing a chord π΄π΅, then the straight line that passes through π and bisects the chord π΄π΅ is perpendicular to π΄π΅. We can say that the measure of angle ππΆπ΅ is 90 degrees. We are told that the length of π΄π is 200 centimeters. And since this is a radius of the circle, π΅π is also 200 centimeters. We are also told that ππΆ is equal to 120 centimeters. Adding these measurements to our diagram, we have a right triangle ππΆπ΅ as shown.

Applying the Pythagorean theorem, the length of π΅πΆ is equal to the square root of 200 squared minus 120 squared. This is equal to 160. As π΅πΆ is equal to 160 centimeters and the point πΆ bisects the line segment π΄π΅, then π΄πΆ is also equal to 160 centimeters. The line segment π΄π΅ is therefore equal to 320 centimeters.

In our next example, we will see how we can apply the theroems to find the area of a triangle.

In the figure below, if ππ΄ is equal to 17.2 centimeters and π΄π΅ is equal to 27.6 centimeters, find the length of the line segment ππΆ and the area of triangle π΄π·π΅ to the nearest tenth.

Since π is the center of the circle and the line segment ππ· bisects the chord π΄π΅ at πΆ, we can apply the chord bisector theorem. This states that if we have a circle with center π containing a chord π΄π΅, then the straight line that passes through π and bisects π΄π΅ is perpendicular to π΄π΅. This means that the measure of angle ππΆπ΄ is 90 degrees. Since the chord π΄π΅ has length 27.6 centimeters and we know that πΆ bisects this chord, π΄πΆ is equal to 27.6 divided by two. This is equal to 13.8 centimeters.

We are also told that the radius ππ΄ is equal to 17.2 centimeters. We can therefore use the Pythagorean theorem in the right triangle ππΆπ΄ such that ππΆ is equal to the square root of 17.2 squared minus 13.8 squared. This is equal to 10.266 and so on. And rounding to the nearest tenth, the line segment ππΆ is equal to 10.3 centimeters.

The second part of our question asks us to calculate the area of triangle π΄π·π΅. Clearing some space, we recall that the area of any triangle is equal to the length of its base multiplied by the length of its perpendicular height divided by two. We know that the base of our triangle π΄π΅ is equal to 27.6 centimeters. The perpendicular height πΆπ· will be equal to the length of ππ· minus the length of ππΆ. ππ· is the radius of the circle, and we know this is equal to 17.2 centimeters. Whilst we could use 10.3 centimeters for ππΆ, it is better for accuracy to use the nonrounded version: ππΆ is equal to 10.266 and so on. Subtracting this from 17.2, we see that the length of πΆπ· is 6.933 and so on centimeters.

We can now calculate the area of triangle π΄π·π΅ by multiplying this by 27.6 centimeters and then dividing by two. This is equal to 95.683 and so on. Once again, we need to round to the nearest tenth, giving us an answer of 95.7 square centimeters.

In our next example, we will see how we can use perpendicular bisectors of chords to find a missing angle.

Line segments π΄π΅ and π΄πΆ are two chords in the circle with center π in two opposite sides of its center, where the measure of angle π΅π΄πΆ is 33 degrees. If π· and πΈ are the midpoints of the line segments π΄π΅ and π΄πΆ, respectively, find the measure of angle π·ππΈ.

We begin by noticing that ππΈ and ππ· both pass through the center of the circle and that they bisect the chords π΄πΆ and π΄π΅, respectively. We can therefore apply the chord bisector theorem, which states if we have a circle with center π containing a chord π΄π΅, then the straight line which passes through π and bisects π΄π΅ is perpendicular to π΄π΅. This means that, on our diagram, the measure of angle ππΈπ΄ and the measure of angle ππ·π΄ are both equal to 90 degrees.

We notice that π΄π·ππΈ is a quadrilateral. And we know that the angles in a quadrilateral sum to 360 degrees. This means that the measure of angle π·ππΈ which we are trying to calculate is equal to 360 minus 90 minus 90 minus 33. This is equal to 147 degrees.

In our final example, we will find the perimeter of a triangle using perpendicular bisectors of chords.

In a circle of center π, π΄π΅ is equal to 35 centimeters, πΆπ΅ is equal to 25 centimeters, and π΄πΆ is equal to 40 centimeters. Given that line segment ππ· is perpendicular to line segment π΅πΆ and line segment ππΈ is perpendicular to line segment π΄πΆ, find the perimeter of triangle πΆπ·πΈ.

We are given in the question the length of the three sides of the triangle πΆπ΅π΄. We know that π΄π΅ is 35 centimeters, πΆπ΅ is 25 centimeters, and π΄πΆ is 40 centimeters. We have been asked to calculate the perimeter of triangle πΆπ·πΈ. We will do this by firstly proving that triangles πΆπ΅π΄ and πΆπ·πΈ are similar using the chord bisector theorem. We notice from the diagram that the line segments ππΈ and ππ· both pass through π and meet the chords π΄πΆ and πΆπ΅ at right angles.

The chord bisector theorem states that if we have a circle with center π containing a chord π΅πΆ, then the straight line that passes through π and is perpendicular to π΅πΆ also bisects π΅πΆ. In our diagram, this means that the length of π΄πΈ is equal to the length πΈπΆ and the length πΆπ· is equal to the length π·π΅.

It is also clear from the diagram that π΄πΆ is equal to two multiplied by πΈπΆ and πΆπ΅ is equal to two multiplied by πΆπ·. As the two triangles πΆπ΅π΄ and πΆπ·πΈ also share the angle πΆ, we have two corresponding sides in proportion and the angle between the two sides is congruent. This proves that the two triangles are similar. And in fact triangle πΆπ΅π΄ is larger than triangle πΆπ·πΈ by a scale factor of two, as the lengths of the corresponding sides are twice as long. Side π΄πΆ is equal to two multiplied by side πΈπΆ, πΆπ΅ is equal to two πΆπ·, and π΄π΅ is equal to two multiplied by πΈπ·.

We can calculate the perimeter of triangle πΆπ΅π΄ by adding 40, 35, and 25. This is equal to 100 centimeters. The perimeter of triangle πΆπ·πΈ will therefore be equal to half of this. This is equal to 50 centimeters.

We have now seen a variety of examples of how perpendicular bisectors of chords can be used to find missing lengths, angle measures, and other unknowns in problems involving circles. We will now recap the key points from this video.

The chord bisector theorem can be summarized in three ways. Firstly, if we have a circle with center π΄ containing a chord π΅πΆ, then the straight line that passes through π΄ and bisects the chord π΅πΆ is perpendicular to π΅πΆ. In the same way, the straight line that passes through π΄ and is perpendicular to π΅πΆ also bisects π΅πΆ. The converse of these two states that the perpendicular bisector of the chord π΅πΆ passes through the center π΄. As already stated, these theorems can be used to find missing lengths, angle measures, and other unknowns in problems involving circles.

## Join Nagwa Classes

Attend live sessions on Nagwa Classes to boost your learning with guidance and advice from an expert teacher!

• Interactive Sessions
• Chat & Messaging
• Realistic Exam Questions