Given that 𝑀𝑋 equals 42 centimeters and 𝑀𝐴 equals 58 centimeters, find the length of line segment 𝐴𝐵.
Let’s start by filling in the measurement information that we were given. So 𝑀𝑋 is 42 centimeters and 𝑀𝐴 is 58 centimeters. We are asked to work out the length of the line segment 𝐴𝐵. We should notice that the line segment 𝐴𝐵 is a chord. That’s because it’s a line segment which joins two distinct points on the circumference. There is also this line 𝑀𝑋, which is a line from the center 𝑀 to the chord. We can also see that there is a right angle of 90 degrees, meaning that this line from the center meets the chord at 90 degrees.
We should recall the property that if we have a circle with center 𝐴 containing a chord, the line segment 𝐵𝐶, then the straight line that passes through 𝐴 and is perpendicular to line segment 𝐵𝐶 also bisects line segment 𝐵𝐶. The application of this property here means that the line segment 𝐴𝑋 is congruent to the line segment 𝑋𝐵. This is because the chord 𝐴𝐵 has been bisected. That means that if we knew the length of either the line segment 𝐴𝑋 or the line segment 𝑋𝐵, then we could double it to find the length of the line segment 𝐴𝐵.
So let’s consider this triangle 𝐴𝑋𝑀. And we know that this is going to be a right triangle because the angle measure of 𝐴𝑋𝑀 will also be 90 degrees. This comes, of course, from the fact that we have a straight line 𝐴𝐵 and the angles on a straight line sum to 180 degrees.
So now we have a right triangle, and we know two of the lengths in this right triangle, and we want to find the length of the third side. So let’s define the length of the line segment 𝐴𝑋 to be 𝑦 centimeters. We can find the value of 𝑦 by applying the Pythagorean theorem. This theorem tells us that in any right triangle, the square of the hypotenuse is equal to the sum of the squares on the other two sides.
And so we have the hypotenuse squared. The hypotenuse is always opposite the right angle. So 58 squared is equal to 42 squared plus 𝑦 squared. 58 squared is equal to 3364, and 42 squared is equal to 1764. Subtracting 1764 from both sides leaves us with 1600 is equal to 𝑦 squared.
Next, we must take the square root of both sides of this equation, remembering that since 𝑦 is a length, then this will be the positive value of the square root. And so we have that 40 is equal to 𝑦. And so we know that the length of the line segment 𝐴𝑋 is 40 centimeters.
We must be careful here. We haven’t finished the question because we were asked for the length of the line segment 𝐴𝐵. This is where our property becomes very important. We know that the line segment 𝐴𝑋 is equal to the line segment 𝐵𝑋. So we can find the length of the line segment 𝐴𝐵 by doubling the length of the line segment 𝐴𝑋. That’s two times 40, which is 80. And so we can give the answer that the length of line segment 𝐴𝐵 is 80 centimeters.