Question Video: Determining Which Chord Is Longer in a Circle Based on the Lengths of Perpendicular Chords Mathematics

Video Transcript

Suppose that 𝐵𝐶 equals eight centimeters and 𝐵𝐴 equals seven centimeters. Which of the following is true? Is it (A) 𝐷𝑀 is equal to 𝑋𝑌, (B) 𝐷𝑀 is greater than 𝑋𝑌, or (C) 𝐷𝑀 is less than 𝑋𝑌?

Let’s begin by adding the lengths 𝐵𝐶 and 𝐵𝐴 to our diagram. These are the distances from the chords 𝐷𝑀 and 𝑋𝑌, respectively, to the center of the circle 𝐵. We recall that the chord that is closer to the center of the circle has a greater length. From the diagram, we see that the chord 𝑋𝑌 is seven centimeters from the center. This is the length of 𝐵𝐴. The chord 𝐷𝑀, on the other hand, is eight centimeters from the center. This means that 𝑋𝑌 is closer to the center of the circle than 𝐷𝑀. As this chord will have a greater length, we can conclude that 𝑋𝑌 is greater than 𝐷𝑀. From the three options listed, the correct answer is option (C) 𝐷𝑀 is less than 𝑋𝑌.