Question Video: Finding a Missing Length Using Equal Chords Mathematics

Video Transcript

Given that 𝐴𝐵 is equal to 𝐶𝐷, 𝑀𝐶 is equal to 10 centimeters, and 𝐷𝐹 is equal to eight centimeters, find the length of line segment 𝑀𝐸.

In this question, we are trying to calculate the length of 𝑀𝐸, which is the distance from the chord 𝐴𝐵 to the center of the circle 𝑀. We begin by recalling that two chords of equal lengths are equidistant from the center. And in this question, we are told that the two chords 𝐴𝐵 and 𝐶𝐷 are equal in length. This means that the length 𝑀𝐹 must be equal to 𝑀𝐸. The line segment 𝑀𝐸 perpendicularly bisects the chord 𝐴𝐵. Likewise, 𝑀𝐹 is the perpendicular bisector of 𝐶𝐷. Since we are told 𝐷𝐹 is equal to eight centimeters, 𝐶𝐹, 𝐴𝐸, and 𝐵𝐸 are all also equal to eight centimeters.

Our next step is to consider the right triangle 𝑀𝐹𝐶. Using the Pythagorean theorem, 𝑀𝐹 squared plus 𝐶𝐹 squared is equal to 𝑀𝐶 squared. By subtracting 𝐶𝐹 squared from both sides and substituting in the values of 𝐶𝐹 and 𝑀𝐶, we have 𝑀𝐹 squared is equal to 10 squared minus eight squared. This is equal to 36. Square rooting both sides of this equation and knowing that 𝑀𝐹 must be a positive answer, we have 𝑀𝐹 is equal to six. Since 𝑀𝐹 is equal to six centimeters, 𝑀𝐸 must also be equal to six centimeters. The perpendicular distance from the center of the chord 𝐴𝐵 to the center of the circle 𝑀, which is the line segment 𝑀𝐸, is equal to six centimeters.