### Video Transcript

Given that π΄π΅ equals 42 centimetres and πΈπΆ equals 10 centimetres in the two concentric circles shown below, find the length of line segment π΄πΆ.

So letβs have a look at the diagram of the circles and the lines and see if we can figure out anything about their properties. Weβre told that the circles are concentric, which means that they share the same centre, in this case, the point π. We can also see that we have some chords in our circles, the chord π·πΆ in the smaller circle and the chord π΄π΅ in the larger circle. A chord is a straight line segment whose endpoints lie on the circle.

Letβs also recall another key fact about chords. And that is that the part perpendicular bisector of a chord passes through the centre of the circle. And a reminder that perpendicular means at 90 degrees. And to bisect means to cut exactly in half. If we look at our diagram, we can see that the line ππΈ cuts the chord π·πΆ and the chord π΄π΅ at 90 degrees. It will also bisect these lines. That means that the line π·πΈ is equal to the line πΈπΆ. And the line π΅πΈ is equal to the line π΄πΈ. So if we fill in the fact that weβre given, that πΈπΆ is 10 centimetres, then the line πΈπ· must also be 10 centimetres. Since weβve established that these lines are the same length.

To find the length of π΄πΆ then, we need to use a fact weβre given that π΄π΅ is 42 centimetres. So if we call our unknown length π₯, then we notice that we still have an unknown length on this line segment π΄π΅. However, recalling that our line ππΈ is a perpendicular bisector, this means that our length π΅πΈ is exactly the same length as the line π΄πΈ. Which means that the length of π΅π· can also be defined as π₯.

Therefore, we can write that π₯ plus 10 plus 10 plus π₯ equals 42. This simplifies to two π₯ plus 20 equals 42. Subtracting 20 from both sides of our equation will give us two π₯ equals 22. And π₯ is equal to 11. Therefore, our final answer is that our line π΄πΆ is 11 centimetres.