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Question Video: Finding the Length of a Chord Using the Properties of Tangents to Circle Mathematics • 11th Grade

Given that the line ๐ด๐ต and the line ๐ด๐ถ are two tangents, find the length of the line segment ๐ต๐ถ.

02:08

Video Transcript

Given that the line ๐ด๐ต and the line ๐ด๐ถ are two tangents, find the length of the line segment ๐ต๐ถ.

So, in the figure, we can see that we have two tangents, ๐ด๐ต and ๐ด๐ถ, which meet at the point ๐ด exterior to the circle. The length of the line segment ๐ด๐ต is 47 centimeters, and the measure of the angle formed where the two tangents intersect is 60 degrees. Weโ€™re asked to find the length of the line segment ๐ต๐ถ, which is the chord connecting points ๐ต and ๐ถ, which are the points where the two tangents intersect the circle.

We can see that we have a triangle formed by the points ๐ด, ๐ต, and ๐ถ. As ๐ด๐ต and ๐ด๐ถ are tangents drawn from the same exterior point to a circle, we can recall a key theorem relating to tangents of circles. It states the following. Given an exterior point to a circle, the lengths of two tangents from the point to the circle are equal. So this tells us that the length of the line segment ๐ด๐ต, which we know to be 47 centimeters, is the same as the length of the line segment ๐ด๐ถ. If the triangle formed by points ๐ด, ๐ต, and ๐ถ has two equal side lengths, then it must be an isosceles triangle, with the base angles ๐ด๐ต๐ถ and ๐ด๐ถ๐ต of equal measure.

But, in fact, as we know that the measure of angle ๐ต๐ด๐ถ is 60 degrees, each of the remaining angles in the triangle must also be of measure 60 degrees. And so the triangle is in fact equilateral. The third side of the triangle must also be of length of 47 centimeters. So, by recalling that tangents drawn from the same exterior point to a circle are equal in length, we found that the length of the line segment ๐ต๐ถ in this figure is 47 centimeters.

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