### Video Transcript

In triangle πππ, where π΄ is the
midpoint of line ππ, what is the name given to the line π΄π? A) Base, B) height, C) hypotenuse,
or D) median.

First, letβs sketch a triangle,
given these conditions. Hereβs a triangle. If we make this side π, weβll
follow the naming convention such that we have π and then π. And this is our triangle
πππ. If π΄ is the midpoint of line ππ,
then π΄ is halfway between π and π. This also means that the segments
ππ΄ and π΄π are equal in length because the midpoint divides the line ππ in
half.

But weβre interested in what we
would call the line π΄π. We know that the base can be any
one of the three sides of the triangle. But the segment π΄π is not one of
the original sides of the triangle. Therefore, it cannot be the
base. What about the height of a
triangle? The height of the triangle depends
on which base youβre using. If we let line ππ be the base,
then this would be the height because the height is the perpendicular distance from
the base to the vertex opposite that base. But remember, weβve just sketched
this triangle. We donβt know that thatβs exactly
what the triangle looks like. So, letβs leave this information
here and keep going.

If we consider the word hypotenuse,
that is the longest side of a right triangle. We donβt know if triangle πππ is
a right triangle. Even if triangle πππ was a right
triangle and π΄ was halfway between π and π, the line ππ΄ would still not be the
hypotenuse because it is a line segment inside the triangle and therefore would not
be the hypotenuse.

So now, we should consider the
definition of a median of a triangle. The median of a triangle is a line
segment joining a vertex to the midpoint of the opposite side. We know that point π΄ is a midpoint
and π is the vertex opposite the line ππ. This means we can say line segment
π΄π is a median. Itβs worth noting that there are
triangles in which the median and the height are the same line. Remember that the height needs to
be perpendicular to the vertex opposite it, if we drew a line that was perpendicular
to the midpoint π΄ and the π vertex fell on that line.

Here is a triangle πππ, where
ππ is the base. Line segment π΄π is still the
median because point π΄ is the midpoint of ππ. But because the line segment π΄π
is perpendicular to the base ππ, in this case, π΄π is also the height. This fact where the median and
height are the same value is true in isosceles triangles. But since we havenβt been told
whether or not triangle πππ is isosceles, the only thing we can say for sure is
that π΄π is the median. We cannot tell if it is the height
or not, which makes option D median the best answer.