# Question Video: Finding the Length of a Segment that Lies between Two Concentric Circles Mathematics • 11th Grade

Given that π΄π΅ = 42 cm and πΈπΆ = 10 cm in the two concentric circles shown below, find the length of line segment π΄πΆ.

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### Video Transcript

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

So letβs have a look at the diagram of the circles and the lines and see if we can figure out anything about their properties. Weβre 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·πΆ 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.

Letβs 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 ππΈ cuts the chord π·πΆ and the chord π΄π΅ at 90 degrees. It will also bisect these lines. That means that the line π·πΈ is equal to the line πΈπΆ. And the line π΅πΈ is equal to the line π΄πΈ. So if we fill in the fact that weβre given, that πΈπΆ is 10 centimetres, then the line πΈπ· must also be 10 centimetres. Since weβve established that these lines are the same length.

To find the length of π΄πΆ then, we need to use a fact weβre 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 ππΈ is a perpendicular bisector, this means that our length π΅πΈ is exactly the same length as the line π΄πΈ. 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΄πΆ is 11 centimetres.