Video Transcript
For the given circle, which of the
following arcs are adjacent? Option (A) the minor arc from 𝐴 to
𝐵 and the minor arc from 𝐶 to 𝐷. Option (B) the minor arc from 𝐴 to
𝐵 and the minor arc from 𝐵 to 𝐶. Option (C) the minor arc from 𝐴 to
𝐷 and the minor arc from 𝐵 to 𝐶. Or option (D) the minor arc from 𝐴
to 𝐶 and the minor arc from 𝐷 to 𝐵.
In this question, we’re given a
circle and we need to determine which of the given pairs of arcs in this circle are
adjacent. To answer this question, let’s
start by recalling what it means for two arcs in our circle to be adjacent. We say that two arcs are adjacent
if they share only a single point or only both endpoints. Therefore, we can determine which
pair of arcs are adjacent by sketching them and determining how many points they
share in common. Let’s start with option (A). We’ll sketch the minor arc from 𝐴
to 𝐵. Remember, there’s two arcs in our
circle from 𝐴 to 𝐵, and we want the shorter one. Next, we’ll sketch the minor arc
from 𝐶 to 𝐷. Once again, this is the shorter
section of the circumference of our circle from 𝐶 to 𝐷. And we can see that these two arcs
share no points in common. So they’re not adjacent.
We can do the same for option
(B). Let’s sketch the minor arc from 𝐴
to 𝐵 and the minor arc from 𝐵 to 𝐶. The minor arc from 𝐴 to 𝐵 is the
shorter section of the circumference of our circle between 𝐴 and 𝐵. And the minor arc from 𝐵 to 𝐶 is
the shorter section of the circumference of our circle between 𝐵 and 𝐶. We can see in our sketch both of
these arcs contain the point 𝐵. In fact, it’s the only point they
share in common. And this means that they’re
adjacent. So the answer to this question is
option (B). We could stop here. However, for due diligence, let’s
check the other two options.
To check option (C), we need to
sketch the minor arc from 𝐴 to 𝐷 and the minor arc from 𝐵 to 𝐶. If we do this, we get the
following. We can see that these two arcs
share no points in common, so they’re not adjacent. Finally, let’s look at option
(D). We need to determine whether the
minor arc from 𝐴 to 𝐶 and the minor arc from 𝐷 to 𝐵 are adjacent. We can sketch both of these arcs
onto our circle, and we notice something interesting. Every single point between 𝐵 and
𝐶 on our circle lies in both arcs. So although they do share a point
in common, they actually share an infinite number of points in common. So these arcs are not adjacent. Therefore, of the given options,
only the minor arc from 𝐴 to 𝐵 and the minor arc from 𝐵 to 𝐶 are adjacent, which
was option (B).
