Video Transcript
Given that 𝐴𝐵 is equal to 𝐶𝐷, which is equal to six 𝑥 plus three centimeters, 𝑀𝐸 is equal to three 𝑥 plus one centimeters, and 𝑀𝑂 equals four centimeters, find the length of line segment 𝐶𝐷.
We are told in the question that the chords 𝐴𝐵 and 𝐶𝐷 are equal in length. They are six 𝑥 plus three centimeters long. We are also told that the perpendicular distance from the center 𝑀 to the chord 𝐴𝐵 is three 𝑥 plus one centimeters and the perpendicular distance from the center 𝑀 to the chord 𝐷𝐶 is four centimeters. We are asked to find the length of the chord 𝐶𝐷. In order to do this, we begin by recalling one of the properties of circles. This states that two chords of equal lengths in the same circle are equidistant from the center.
In this question, since the chords 𝐴𝐵 and 𝐶𝐷 are equal in length, the line segments 𝑀𝑂 and 𝑀𝐸 must also be equal in length. Three 𝑥 plus one centimeters must therefore be equal to four centimeters. We can solve the equation three 𝑥 plus one equals four by firstly subtracting one from both sides. Dividing through by three, we have a value of 𝑥 equal to one. As we need to calculate the length of 𝐶𝐷, we can substitute this value of 𝑥 into the expression six 𝑥 plus three. This is equal to six multiplied by one plus three, which in turn is equal to nine.
We can therefore conclude that the line segment 𝐶𝐷 has length nine centimeters.
