### Video Transcript

In the shown figure, π΄π equals 7.5 centimetres, π΅π equals 14.5 centimetres, and π΄πΆ equals 20 centimetres. Given that all sides of triangle π΄π΅πΆ are tangent to the shown circle, determine the length of π΅πΆ.

Letβs begin by adding the length that weβre given in the question to the diagram. So we have the lengths of two of the sides of this triangle and weβre looking to determine the length of π΅πΆ, which is the third side. Weβre also given the key piece of information that all three sides of the triangle π΄π΅πΆ are tangent to the shown circle. Letβs think about what we know about the lengths of tangents drawn from exterior points to circles.

Hereβs an important fact. If two segments from the same exterior point are tangent to a circle, then they are congruent. In practice, what this means is that the two line segments drawn from the point π΄ to the circle are both equal in length. The same is true for the segments drawn from π΅ and those drawn from πΆ. How would this help us with answering the question? Well, remember we want to determine the length of π΅πΆ. So in our diagram, we need to know the length of the pink segment and the length of the green segment.

Using the result we just discussed, we actually already know the length of the pink segment. Itβs equal to π΅π which is 14.5 centimetres. So we need to think about how weβre going to calculate the length of the green segment. And to do this, Iβm going to add in a couple of labels on the other two sides of this triangle. So just as we have the point π on the side π΄π΅, we now have the point π on the side π΄πΆ and the point π on the side π΅πΆ, which are the points where these tangents touch the circle.

We know the full length of the side π΄πΆ. But we want to know how much of this is due to the orange part π΄π and how much is due to the green part ππΆ. Well, applying the same result again, we know that the line segment π΄π is congruent to π΄π. And therefore, itβs equal to 7.5 centimetres. The line segment πΆπ can therefore be found by subtracting π΄π from the length of π΄πΆ: 20 minus 7.5. So we know that πΆπ is 12.5 centimetres.

Applying our key result a third time, we know that the two segments drawn from the point πΆ are congruent to each other. And therefore, πΆπ is congruent to πΆπ. πΆπ is 12.5 centimetres. So our final step in this problem, we need to determine the length of π΅πΆ by summing the two segments π΅π and πΆπ. So π΅πΆ is equal to 14.5 plus 12.5. Itβs 27 centimetres.

The key result which we applied three times within this question is that if two segments from the same exterior point are tangent to a circle, then they are congruent β meaning theyβre equal in length.