Given that the line 𝐴𝐵 and the line 𝐴𝐶 are two tangents, find the length of the line segment 𝐵𝐶.
So, in the figure, we can see that we have two tangents, 𝐴𝐵 and 𝐴𝐶, which meet at the point 𝐴 exterior to the circle. The length of the line segment 𝐴𝐵 is 47 centimeters, and the measure of the angle formed where the two tangents intersect is 60 degrees. We’re asked to find the length of the line segment 𝐵𝐶, which is the chord connecting points 𝐵 and 𝐶, which are the points where the two tangents intersect the circle.
We can see that we have a triangle formed by the points 𝐴, 𝐵, and 𝐶. As 𝐴𝐵 and 𝐴𝐶 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 𝐴𝐵, which we know to be 47 centimeters, is the same as the length of the line segment 𝐴𝐶. If the triangle formed by points 𝐴, 𝐵, and 𝐶 has two equal side lengths, then it must be an isosceles triangle, with the base angles 𝐴𝐵𝐶 and 𝐴𝐶𝐵 of equal measure.
But, in fact, as we know that the measure of angle 𝐵𝐴𝐶 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 𝐵𝐶 in this figure is 47 centimeters.