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

Given that ๐ด๐ต = 42 cm and ๐ธ๐ถ = 10 cm in the two concentric circles shown below, find the length of line segment ๐ด๐ถ.


Video Transcript

Given that ๐ด๐ต equals 42 centimetres and ๐ธ๐ถ equals 10 centimetres in the two concentric circles shown below, find the length of line segment ๐ด๐ถ.

So letโ€™s have a look at the diagram of the circles and the lines and see if we can figure out anything about their properties. Weโ€™re told that the circles are concentric, which means that they share the same centre, in this case, the point ๐‘‚. We can also see that we have some chords in our circles, the chord ๐ท๐ถ in the smaller circle and the chord ๐ด๐ต in the larger circle. A chord is a straight line segment whose endpoints lie on the circle.

Letโ€™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 ๐‘‚๐ธ cuts the chord ๐ท๐ถ and the chord ๐ด๐ต at 90 degrees. It will also bisect these lines. That means that the line ๐ท๐ธ is equal to the line ๐ธ๐ถ. And the line ๐ต๐ธ is equal to the line ๐ด๐ธ. So if we fill in the fact that weโ€™re given, that ๐ธ๐ถ is 10 centimetres, then the line ๐ธ๐ท must also be 10 centimetres. Since weโ€™ve established that these lines are the same length.

To find the length of ๐ด๐ถ then, we need to use a fact weโ€™re given that ๐ด๐ต is 42 centimetres. So if we call our unknown length ๐‘ฅ, then we notice that we still have an unknown length on this line segment ๐ด๐ต. However, recalling that our line ๐‘‚๐ธ is a perpendicular bisector, this means that our length ๐ต๐ธ is exactly the same length as the line ๐ด๐ธ. Which means that the length of ๐ต๐ท can also be defined as ๐‘ฅ.

Therefore, we can write that ๐‘ฅ plus 10 plus 10 plus ๐‘ฅ equals 42. This simplifies to two ๐‘ฅ plus 20 equals 42. Subtracting 20 from both sides of our equation will give us two ๐‘ฅ equals 22. And ๐‘ฅ is equal to 11. Therefore, our final answer is that our line ๐ด๐ถ is 11 centimetres.

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