Video Transcript
Suppose that 𝐷𝐶 equals 50
centimeters and 𝐴𝐵 equals 52 centimeters. Which of the following is true? Is it (A) 𝑀𝑋 is less than 𝑀𝑌,
(B) 𝑀𝑋 is greater than 𝑀𝑌, or (C) 𝑀𝑋 equals 𝑀𝑌?
Let’s begin by adding any of the
lengths given to our diagram. We’re told that 𝐷𝐶 is 50
centimeters. That’s the length of the line
segment between points 𝐷 and 𝐶 on the circumference of the circle. We’re also told that 𝐴𝐵, the
length of the line segment that joins two other points on the circumference of the
circle, is 52 centimeters. The diagram also shows each of
these lines with respect to the center of the circle. Both 𝐷𝐶 and 𝐴𝐵 are chords
within the circle. And so the radii that were to pass
through 𝑀𝑌 and 𝑀𝑋, respectively, could be constructed as perpendicular bisectors
of each of these chords.
Now, the question wants us to
compare line segment 𝑀𝑌 with line segment 𝑀𝑋. So let’s think about what we know
about two chords in a circle based on their distance from the center, and vice
versa. We know that when we have a pair of
chords constructed within a circle, the chord which is closer to the center of the
circle will have a greater length. Now, comparing the two chords we’ve
been given, 𝐴𝐵 and 𝐷𝐶, we see that 𝐴𝐵 is two centimeters longer than 𝐷𝐶. So 𝐴𝐵 is longer in length than
𝐷𝐶. This means then that chord 𝐴𝐵
must be closer to the center of the circle, that’s point 𝑀, than the chord
𝐷𝐶. If this is the case, then point 𝑋
must be closer to point 𝑀 than 𝑌 must be. So we can see that 𝑀𝑋 must be
less than 𝑀𝑌.
So, since 𝐴𝐵 is longer in length
than 𝐷𝐶, 𝐴𝐵 is closer to the center of the circle, which means that 𝑀𝑋 is less
than 𝑀𝑌. The correct answer is (A).
