Question Video: Finding the Length of a Chord in a Circle | Nagwa Question Video: Finding the Length of a Chord in a Circle | Nagwa

# Question Video: Finding the Length of a Chord in a Circle Mathematics • Third Year of Preparatory School

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Given π΄π = 200 cm and ππΆ = 120 cm, find the line segment π΄π΅.

03:43

### Video Transcript

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

So, weβve been given a diagram of a circle and then some internal line segments. Weβre also told that π΄π is 200 centimeters and ππΆ is 120 centimeters. Now, π΄π wasnβt already drawn on the diagram, so we can go ahead and add that line in and label it with its length. We can also include the length of ππΆ. Itβs 120 centimeters. The length weβre looking to calculate is the length of the line segment π΄π΅, which we can see is a cord of this circle.

Now, a key point on the diagram is that these little marks have been added to the line segments π΄πΆ and πΆπ΅, which indicate that they are the same length. This means that the line segment ππΆ, or ππ· if we go all the way to the circumference of the circle, bisects the line segment π΄π΅ as itβs divided it up into two equal parts. This further tells us that the angle where these two lines meet must be a right angle because if a line drawn from the center of a circle to the circumference divides a cord in half, then it will be a perpendicular bisector of that cord.

So, we know that ππ· is the perpendicular bisector of π΄π΅. And by drawing in this right angle, we see that we now have a right triangle π΄ππΆ in which we know two lengths. We can, therefore, apply the Pythagorean theorem to find the third length π΄πΆ. And as π΄πΆ is half of π΄π΅, weβll be able to double this value to find the total length of π΄π΅.

The Pythagorean theorem tells us that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two shorter sides. So, if the two shorter sides are labelled as π and π, and the hypotenuse is labeled as π, we have the equation π squared plus π squared is equal to π squared.

In our right triangle π΄ππΆ, the two shorter sides are ππΆ, which is 120, and π΄πΆ, which we donβt know, and the hypotenuse is 200. So, by applying the Pythagorean theorem, we have the equation 120 squared plus π΄πΆ squared is equal to 200 squared. We can evaluate both 120 squared and 200 squared and then subtract 14400 from each side of the equation, giving π΄πΆ squared equals 25600. To solve for π΄πΆ, we square root both sides of the equation, taking only the positive square root as π΄πΆ is a length. We find that π΄πΆ is equal to 160.

Finally then, recall that we were asked to find the length of the line segment π΄π΅. And as π΄πΆ is half of π΄π΅, we can find π΄π΅ by doubling the value weβve just found. The length of π΄π΅ is, therefore, two multiplied by 160, which is 320. Including the appropriate units then, we have that the length of the line segment π΄π΅ is 320 centimeters.

Itβs also worth pointing out that we could have worked in triangle π΅πΆπ rather than triangle π΄πΆπ. The Line ππ΅ is also a radius of the circle just like the line ππ΄. And so, these two triangles are congruent to one another. We could have applied the Pythagorean theorem in this triangle. And it would have given exactly the same result.

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