Question Video: Finding the Lengths of Two Line Segments Using Properties of Circle Tangents Mathematics

Video Transcript

Determine 𝐴𝑀 and 𝐴𝐵, rounding to the nearest hundredth.

And then we see we have a circle center 𝑀 with point 𝐴 that lies outside the circle. Let’s inspect some of the lines in this circle. We have a pair of lines that join the center to the circle to a point on its circumference. Those are line segments 𝑀𝐵 and 𝑀𝐶. These lines must therefore be the radii of the circle. Since 𝑀𝐵 is six centimeters and all radii have the same length, then 𝑀𝐶 must also be six centimeters.

Then, we have two line segments that join a point on the circumference to a point outside the circle. These lines are portions of tangents to the circle. So we call them tangent segments. 𝐵𝐴 and 𝐶𝐴 are tangent segments. And so we recall what we know about a pair of tangent segments that meet at a point outside the circle. They’re said to be congruent. In other words, they are equal in length. We’re told that line segment 𝐶𝐴 is 10.73 centimeters. And so this means that line segment 𝐴𝐵 must also be 10.73 centimeters. And in fact that’s really useful because we’ve already answered part of the question. We’ve determined the length of line segment 𝐴𝐵. It’s 10.73 centimeters.

So, with all of this on our diagram, how do we determine the length of line segment 𝐴𝑀? That’s this line segment here that joins a point at the center to the point where our two tangent segments meet. Well, to answer this part of the question, we remember that a radius and a tangent meet at an angle of 90 degrees. So angle 𝐴𝐶𝑀 and angle 𝐴𝐵𝑀 are both equal to 90 degrees. And then we observe that both triangles 𝐴𝐶𝑀 and 𝐴𝐵𝑀 are right triangles.

We can therefore use either one of these triangles to find the length 𝐴𝑀. Let’s use triangle 𝐴𝐶𝑀. But of course they’re congruent, so the maths is going to be the same. Since we have a right triangle for which we know two of its lengths, we can find the third length using the Pythagorean theorem, sometimes shortened to 𝑎 squared plus 𝑏 squared equals 𝑐 squared, where 𝑐 is the hypotenuse and 𝑎 and 𝑏 are the other two lengths. The Pythagorean theorem tells us that the sum of the squares of the two shorter sides is equal to the square of the hypotenuse.

𝑀𝐴 is the hypotenuse of this triangle. It’s the longest side, which is the side directly opposite the right angle. So let’s define this to be equal to 𝑐 centimeters. Then, substituting what we know about our triangle into the formula for the Pythagorean theorem, and we get six squared plus 10.73 squared equals 𝑐 squared. Evaluating the left-hand side gives us 151.1329. So how do we solve this equation for 𝑐?

Well, we take the square root of both sides of our equation. Now, generally, when we do this, we take the positive and negative square root of the numerical part. But 𝑐 is a length. It cannot be negative. And so we only take the positive square root in this case. The square root of 151.1329 is 12.2936 and so on. Correct to two decimal places, which is equivalent to the nearest hundredth, we get 12.29.

And so we found the length of line segments 𝐴𝑀 and 𝐴𝐵. They are 12.29 centimeters and 10.73 centimeters, respectively.