Given a circle 𝑀 with two chords
𝐴𝐷 and 𝐵𝐶 that have equal lengths and the arc from 𝐴 to 𝐷 with a length of
five centimeters, what is the length of the arc from 𝐵 to 𝐶?
We’re given two chords in a circle
with equal lengths, 𝐴𝐷 and 𝐵𝐶. We can highlight these chords on
our diagram and the fact that they have equal length. We’re also told that the length of
the minor arc from 𝐴 to 𝐷 is five centimeters. We can also add this to our
diagram. We need to use this to determine
the length of the minor arc from 𝐵 to 𝐶. We can answer this question
geometrically by noticing 𝐴𝑀, 𝐵𝑀, 𝐶𝑀, and 𝐷𝑀 are radii of the circle, so
they have equal length. This means that triangles 𝐴𝑀𝐷
and 𝐵𝑀𝐶 are congruent. So the measures of the central
angles of these two arcs are equal.
Then, since the central angles of
these two arcs are equal, the lengths are equal, meaning that 𝐵𝐶 has length five
centimeters. However, we could’ve also answered
this question by just recalling if the chords between two points on a circle are
equal, then their arc lengths are also equal. Using either method, we were able
to show the length of the minor arc from 𝐵 to 𝐶 is five centimeters.