Question Video: Finding the Length of Two Segments Drawn in a Circle Using the Ratio between Them Mathematics • 11th Grade

If ๐ธ๐ด/๐ธ๐ต = 8/7, ๐ธ๐ถ = 7 cm, and ๐ธ๐ท = 8 cm, find the lengths of line segment ๐ธ๐ต and line segment ๐ธ๐ด.

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Video Transcript

If ๐ธ๐ด over ๐ธ๐ต is equal to eight over seven, ๐ธ๐ถ equals seven centimeters, and ๐ธ๐ท equals eight centimeters, find the lengths of line segment ๐ธ๐ต and line segment ๐ธ๐ด.

First, letโ€™s fill in what we know. ๐ธ๐ถ equals seven centimeters. ๐ธ๐ท equals eight centimeters. We have to think carefully about how we would note ๐ธ๐ด over ๐ธ๐ต. This is a ratio relationship. One way we could note this is by saying ๐ธ๐ต is equal to seven ๐‘ฅ and ๐ธ๐ด is equal to eight ๐‘ฅ. For example, if ๐ธ๐ด was equal to 16, ๐ธ๐ต would have to be equal to 14 because 14 to 16 is in the ratio of seven to eight.

So, for now, weโ€™ll leave it as seven ๐‘ฅ to eight ๐‘ฅ. Thereโ€™s something else we need to remember about chords to solve this problem. If two chords intersect, we end up with four segments. And the product of the two segments in each chord must be equal to the product of the two segments in the other chord. Here, we have ๐‘Ž times ๐‘ must be equal to ๐‘ times ๐‘‘.

And in our problem, that means seven centimeters times eight centimeters must be equal to seven ๐‘ฅ times eight ๐‘ฅ. Seven times eight is 56. So, we have 56 centimeters squared on the left. Seven times eight is 56 and ๐‘ฅ times ๐‘ฅ is ๐‘ฅ squared. From there, we divide both sides by 56. 56 divided by 56 is one. If one centimeter squared is equal to ๐‘ฅ squared and we take the square root of both sides, the square root of one centimeter squared would be one centimeter and the square root of ๐‘ฅ would be ๐‘ฅ.

Since ๐‘ฅ equals one centimeter, we multiply seven times one centimeter to get seven centimeters. And eight times one centimeter equals eight centimeters. So, ๐ธ๐ต equals seven centimeters and ๐ธ๐ด equals eight centimeters.

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