### Video Transcript

Given π΄π equals 200 centimeters and ππΆ equals 120 centimeters, find the length of the line segment π΄π΅.

So, weβve been given a diagram of a circle and then some internal line segments. Weβre also told that π΄π is 200 centimeters and ππΆ is 120 centimeters. Now, π΄π wasnβt 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 ππΆ. Itβs 120 centimeters. The length weβre 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΄πΆ and πΆπ΅, which indicate that they are the same length. This means that the line segment ππΆ, or ππ· if we go all the way to the circumference of the circle, bisects the line segment π΄π΅ as itβs 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 ππ· is the perpendicular bisector of π΄π΅. And by drawing in this right angle, we see that we now have a right triangle π΄ππΆ in which we know two lengths. We can, therefore, apply the Pythagorean theorem to find the third length π΄πΆ. And as π΄πΆ is half of π΄π΅, weβll 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΄ππΆ, the two shorter sides are ππΆ, which is 120, and π΄πΆ, which we donβt know, and the hypotenuse is 200. So, by applying the Pythagorean theorem, we have the equation 120 squared plus π΄πΆ 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΄πΆ squared equals 25600. To solve for π΄πΆ, we square root both sides of the equation, taking only the positive square root as π΄πΆ is a length. We find that π΄πΆ is equal to 160.

Finally then, recall that we were asked to find the length of the line segment π΄π΅. And as π΄πΆ is half of π΄π΅, we can find π΄π΅ by doubling the value weβve 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.

Itβs also worth pointing out that we could have worked in triangle π΅πΆπ rather than triangle π΄πΆπ. The Line ππ΅ is also a radius of the circle just like the line ππ΄. 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.