Given 𝐴𝑀 equals 200 centimeters and 𝑀𝐶 equals 120 centimeters, find the length of the line segment 𝐴𝐵.
So, we’ve been given a diagram of a circle and then some internal line segments. We’re also told that 𝐴𝑀 is 200 centimeters and 𝑀𝐶 is 120 centimeters. Now, 𝐴𝑀 wasn’t already drawn on the diagram, so we can go ahead and add that line in and label it with its length. We can also include the length of 𝑀𝐶. It’s 120 centimeters. The length we’re looking to calculate is the length of the line segment 𝐴𝐵, which we can see is a cord of this circle.
Now, a key point on the diagram is that these little marks have been added to the line segments 𝐴𝐶 and 𝐶𝐵, which indicate that they are the same length. This means that the line segment 𝑀𝐶, or 𝑀𝐷 if we go all the way to the circumference of the circle, bisects the line segment 𝐴𝐵 as it’s divided it up into two equal parts. This further tells us that the angle where these two lines meet must be a right angle because if a line drawn from the center of a circle to the circumference divides a cord in half, then it will be a perpendicular bisector of that cord.
So, we know that 𝑀𝐷 is the perpendicular bisector of 𝐴𝐵. And by drawing in this right angle, we see that we now have a right triangle 𝐴𝑀𝐶 in which we know two lengths. We can, therefore, apply the Pythagorean theorem to find the third length 𝐴𝐶. And as 𝐴𝐶 is half of 𝐴𝐵, we’ll be able to double this value to find the total length of 𝐴𝐵.
The Pythagorean theorem tells us that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two shorter sides. So, if the two shorter sides are labelled as 𝑎 and 𝑏, and the hypotenuse is labeled as 𝑐, we have the equation 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared.
In our right triangle 𝐴𝑀𝐶, the two shorter sides are 𝑀𝐶, which is 120, and 𝐴𝐶, which we don’t know, and the hypotenuse is 200. So, by applying the Pythagorean theorem, we have the equation 120 squared plus 𝐴𝐶 squared is equal to 200 squared. We can evaluate both 120 squared and 200 squared and then subtract 14400 from each side of the equation, giving 𝐴𝐶 squared equals 25600. To solve for 𝐴𝐶, we square root both sides of the equation, taking only the positive square root as 𝐴𝐶 is a length. We find that 𝐴𝐶 is equal to 160.
Finally then, recall that we were asked to find the length of the line segment 𝐴𝐵. And as 𝐴𝐶 is half of 𝐴𝐵, we can find 𝐴𝐵 by doubling the value we’ve just found. The length of 𝐴𝐵 is, therefore, two multiplied by 160, which is 320. Including the appropriate units then, we have that the length of the line segment 𝐴𝐵 is 320 centimeters.
It’s also worth pointing out that we could have worked in triangle 𝐵𝐶𝑀 rather than triangle 𝐴𝐶𝑀. The Line 𝑀𝐵 is also a radius of the circle just like the line 𝑀𝐴. And so, these two triangles are congruent to one another. We could have applied the Pythagorean theorem in this triangle. And it would have given exactly the same result.