Question Video: Finding the Length of a Chord in a Circle Mathematics

Given that ๐ธ๐ถ = 4, ๐ธ๐ท = 15, and ๐ธ๐ต = 6, find the length of the line segment ๐ธ๐ด.

02:06

Video Transcript

Given that ๐ธ๐ถ equals four, ๐ธ๐ท equals 15, and ๐ธ๐ต equals six, find the length of the line segment ๐ธ๐ด.

Letโ€™s look at the diagram more closely. We can see that it consists of a circle, in which there are two intersecting chords. Theyโ€™re the lines ๐ด๐ต and ๐ถ๐ท. Weโ€™ve also been given various lengths that we can add to our diagram. The length of the line segment ๐ธ๐ถ is four. The length of the line segment ๐ธ๐ท is 15. And the length of the line segment ๐ธ๐ต is six.

Weโ€™re asked to work out the length of the line segment ๐ธ๐ด. So we need to recall the relationship that exists between the lengths of the line segments of intersecting chords. We remember that โ€œif two chords intersect in a circle, then the products of the lengths of the chord segments are equal.โ€

Our two chords intersect inside the circle at the point ๐ธ. So we have that the product of the lengths of the chord segments of the orange chord, thatโ€™s ๐ธ๐ด multiplied by ๐ธ๐ต, is equal to the product of the length of the chord segment of the pink cord. Thatโ€™s ๐ธ๐ถ multiplied by ๐ธ๐ท. We know the lengths of the chord segments ๐ธ๐ต, ๐ธ๐ถ, and ๐ธ๐ท. So we can substitute their values into this equation given that the length of the chord segment ๐ธ๐ด multiplied by six is equal to four multiplied by 15.

Dividing both sides of this equation through by six gives a calculation that we can use to find the length of the chord segment ๐ธ๐ด. Itโ€™s equal to four multiplied by 15 over six. Four multiplied by 15 is equal to 60. And 60 divided by six is equal to 10.

Using then the relationship between the lengths of chord segments for chords which intersect inside a circle, we found that the length of the chord segment ๐ธ๐ด is 10. Thereโ€™re no units for this as there were no units for the original measurements we were given for the other three chord segments.

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