Question Video: Finding the Length of a Chord Using the Chords Theorem Mathematics • 11th Grade

Given that the points ๐ด, ๐ต, ๐ถ, and ๐ท lie on a circle, find the length of segment ๐ต๐ด.

01:59

Video Transcript

Given that the points ๐ด, ๐ต, ๐ถ, and ๐ท lie on a circle, find the length of segment ๐ต๐ด.

So weโ€™d like to find this length of the segment. And we are also told that ๐ด, ๐ต, ๐ถ, and ๐ท lie on a circle, so maybe something like this. Since we have two lines containing the two segments ๐ต๐ด and ๐ถ๐ท intersecting at point ๐ธ, we can use this to solve for the length of ๐ธ๐ด. And solving for the length of ๐ธ๐ด will help us find the length of ๐ต๐ด.

To do so, we can use the fact that ๐ธ๐ด times ๐ธ๐ต will be equal to ๐ธ๐ถ times ๐ธ๐ท. And again, finding this length ๐ธ๐ด is whatโ€™s going to help us find the length of ๐ต๐ด. So we can go ahead and write ๐ธ๐ด because we donโ€™t know what it is. And the length of ๐ธ๐ต is 36. The length ๐ธ๐ถ we can actually find, because all we need to do is take 39 centimetres plus 33 centimetres to give us 72 centimetres. Thatโ€™s the length of ๐ธ๐ถ. And then, lastly, we need to multiply by ๐ธ๐ท, which is 33 centimetres.

So we can begin solving for ๐ธ๐ด by multiplying 72 and 33 on the right-hand side of the equation. 72 times 33 is 2367. So now to solve for ๐ธ๐ด, we divide both sides of the equation by 36. And we find that the length of segment ๐ธ๐ด is 66 centimetres.

And now we can use this to find the length of ๐ต๐ด. And this is because the length of ๐ต๐ด will be equal to ๐ธ๐ด minus ๐ธ๐ต. ๐ธ๐ด we found to be 66 centimetres. ๐ธ๐ต is 36 centimetres. So subtracting those, we find that the length of segment ๐ต๐ด would be 30 centimetres.

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