Video Transcript
For the given circle, which of the
following arcs are adjacent? Option (A) the minor arc from ๐ด to
๐ต and the minor arc from ๐ถ to ๐ท. Option (B) the minor arc from ๐ด to
๐ต and the minor arc from ๐ต to ๐ถ. Option (C) the minor arc from ๐ด to
๐ท and the minor arc from ๐ต to ๐ถ. Or option (D) the minor arc from ๐ด
to ๐ถ and the minor arc from ๐ท to ๐ต.
In this question, weโre given a
circle and we need to determine which of the given pairs of arcs in this circle are
adjacent. To answer this question, letโs
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. Letโs start with option (A). Weโll sketch the minor arc from ๐ด
to ๐ต. Remember, thereโs two arcs in our
circle from ๐ด to ๐ต, and we want the shorter one. Next, weโll sketch the minor arc
from ๐ถ to ๐ท. Once again, this is the shorter
section of the circumference of our circle from ๐ถ to ๐ท. And we can see that these two arcs
share no points in common. So theyโre not adjacent.
We can do the same for option
(B). Letโs sketch the minor arc from ๐ด
to ๐ต and the minor arc from ๐ต to ๐ถ. The minor arc from ๐ด to ๐ต is the
shorter section of the circumference of our circle between ๐ด and ๐ต. And the minor arc from ๐ต to ๐ถ is
the shorter section of the circumference of our circle between ๐ต and ๐ถ. We can see in our sketch both of
these arcs contain the point ๐ต. In fact, itโs the only point they
share in common. And this means that theyโre
adjacent. So the answer to this question is
option (B). We could stop here. However, for due diligence, letโs
check the other two options.
To check option (C), we need to
sketch the minor arc from ๐ด to ๐ท and the minor arc from ๐ต to ๐ถ. If we do this, we get the
following. We can see that these two arcs
share no points in common, so theyโre not adjacent. Finally, letโs look at option
(D). We need to determine whether the
minor arc from ๐ด to ๐ถ and the minor arc from ๐ท to ๐ต are adjacent. We can sketch both of these arcs
onto our circle, and we notice something interesting. Every single point between ๐ต and
๐ถ 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 ๐ด to ๐ต and the minor arc from ๐ต to ๐ถ are adjacent, which
was option (B).