Video Transcript
Points 𝑋 and 𝑌 are midpoints of
segments 𝐴𝐵 and 𝐶𝐷, respectively, and point 𝑀 is at the center of the
circle. If 𝐴𝐵 equals 60, what is
𝐶𝑌?
We can begin by highlighting that
we’re told that 𝑋 and 𝑌 are the midpoints of segments 𝐴𝐵 and 𝐶𝐷,
respectively. Line segments 𝐴𝐵 and 𝐶𝐷 are in
fact chords of the circle because in each line segment, the endpoints both lie on
the circle. The point 𝑀 is the center of the
circle.
Now, consider that since 𝑋 and 𝑌
are midpoints of chords 𝐴𝐵 and 𝐶𝐷, respectively, then we can say that they have
been bisected by points 𝑋 and 𝑌. Since line segments 𝑋𝑀 and 𝑌𝑀
pass through the center 𝑀, then we recall that this means that 𝑋𝑀 and 𝑌𝑀 are
both perpendicular bisectors of the respective chords 𝐴𝐵 and 𝐶𝐷. From the markings on the diagram,
we can observe that the line segments 𝑋𝑀 and 𝑌𝑀 are marked as congruent. This means that the distance from
the center 𝑀 to the chord 𝐴𝐵 is the same as the distance from 𝑀 to the chord
𝐶𝐷. We can say then that the chords are
equidistant from the center.
Let’s consider for a moment why
these distances have to be the perpendicular distance from the chord to the center,
by taking a different circle with center 𝑃. Just because we can draw two chords
and draw any two congruent line segments from each chord to the center would not
mean that the chords are equidistant. For example, these two pink chords
are clearly not equidistant from the center. We must use the perpendicular
distance from the chords to the center to conclude if the chords are equidistant or
not. And we have already demonstrated
that in our circle with center 𝑀, we have two perpendicular line segments.
We can then recall the equidistant
chords theorem. This theorem tells us that if two
chords in the same circle are equidistant from the center of the circle, then their
lengths are equal. So, because chords 𝐴𝐵 and 𝐶𝐷
are equidistant from the center 𝑀, then we can write that 𝐴𝐵 equals 𝐶𝐷. We are given that 𝐴𝐵 is equal to
60, or 60 length units. Therefore, we know that 𝐶𝐷 must
also be 60 length units. We need to find the length of 𝐶𝑌,
but we know that 𝑌 is the midpoint of 𝐶𝐷. And that means the length of 𝐶𝑌
must be half of 60, which is 30.
We can therefore give the answer
that 𝐶𝑌 is 30, and if we needed to give units, they would be length units.
