Video Transcript
Line 𝐴𝐶 is tangent to a circle of center 𝑀 at the point 𝐴. Given that 𝐵𝑀 equals 55 centimeters and 𝐴𝐶 equals 96 centimeters, what is 𝐵𝐶?
We can begin by labeling our diagram with the information we know. 𝐴𝐶 equals 96 centimeters. 𝐵𝑀 equals 55 centimeters. Since point 𝑀 is the center of the circle and 𝐵 is on the outside of the circle, we know that 𝐵𝑀 is a radius of this circle. From 𝑀 to 𝐴 would be an additional radius of this circle, which makes 𝑀𝐴 also 55 centimeters. The unknown we’re looking for is segment 𝐵𝐶. Let’s label that as 𝑥 centimeters.
The key to solving this is recognizing that line 𝐴𝐶 is a tangent to our circle. And a radius and a tangent are perpendicular if they intersect at the point of tangency. 𝑀𝐴 intersects line 𝐴𝐶 at the point of tangency, point 𝐴. Therefore, angle 𝐵𝐴𝐶 equals 90 degrees. And 𝐴𝐵𝐶 is a right triangle. We can find the length of 𝐵𝐶 using the Pythagorean theorem. Our unknown value 𝑥 is the hypotenuse. So we can say 𝑥 squared equals 96 squared plus 110 squared. Remember, we need the distance from 𝐵 to 𝐴, which means we’ll have to add 55 plus 55 to get 110 here.
96 squared plus 110 squared equals 21316. Taking the square root of both sides, we find that 𝑥 equals the square root of 21316, which is 146. Remember, this is a distance, so we’re only interested in the positive square root. Plugging that into our diagram, we see that the measure of 𝐵𝐶 equals 146 centimeters.
