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

Suppose that 𝐡𝐢 = 8 cm and 𝐡𝐴 = 7 cm. Which of the following is true? [A] 𝐷𝑀 = π‘‹π‘Œ [B] 𝐷𝑀 > π‘‹π‘Œ [C] 𝐷𝑀 < π‘‹π‘Œ

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Video Transcript

Suppose that 𝐡𝐢 equals eight centimeters and 𝐡𝐴 equals seven centimeters. Which of the following is true? Is it (A) 𝐷𝑀 is equal to π‘‹π‘Œ, (B) 𝐷𝑀 is greater than π‘‹π‘Œ, or (C) 𝐷𝑀 is less than π‘‹π‘Œ?

Let’s begin by adding the lengths 𝐡𝐢 and 𝐡𝐴 to our diagram. These are the distances from the chords 𝐷𝑀 and π‘‹π‘Œ, respectively, to the center of the circle 𝐡. 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 π‘‹π‘Œ is seven centimeters from the center. This is the length of 𝐡𝐴. The chord 𝐷𝑀, on the other hand, is eight centimeters from the center. This means that π‘‹π‘Œ is closer to the center of the circle than 𝐷𝑀. As this chord will have a greater length, we can conclude that π‘‹π‘Œ is greater than 𝐷𝑀. From the three options listed, the correct answer is option (C) 𝐷𝑀 is less than π‘‹π‘Œ.

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