Video Transcript
In triangle 𝑋𝑌𝑍, where 𝐴 is the
midpoint of line 𝑋𝑌, what is the name given to the line 𝐴𝑍? A) Base, B) height, C) hypotenuse,
or D) median.
First, let’s sketch a triangle,
given these conditions. Here’s a triangle. If we make this side 𝑋, we’ll
follow the naming convention such that we have 𝑌 and then 𝑍. And this is our triangle
𝑋𝑌𝑍. If 𝐴 is the midpoint of line 𝑋𝑌,
then 𝐴 is halfway between 𝑋 and 𝑌. This also means that the segments
𝑌𝐴 and 𝐴𝑋 are equal in length because the midpoint divides the line 𝑋𝑌 in
half.
But we’re interested in what we
would call the line 𝐴𝑍. We know that the base can be any
one of the three sides of the triangle. But the segment 𝐴𝑍 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 you’re using. If we let line 𝑋𝑌 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, we’ve just sketched
this triangle. We don’t know that that’s exactly
what the triangle looks like. So, let’s leave this information
here and keep going.
If we consider the word hypotenuse,
that is the longest side of a right triangle. We don’t know if triangle 𝑋𝑌𝑍 is
a right triangle. Even if triangle 𝑋𝑌𝑍 was a right
triangle and 𝐴 was halfway between 𝑋 and 𝑌, the line 𝑍𝐴 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 𝐴 is a midpoint
and 𝑍 is the vertex opposite the line 𝑋𝑌. This means we can say line segment
𝐴𝑍 is a median. It’s 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 𝐴 and the 𝑍 vertex fell on that line.
Here is a triangle 𝑋𝑌𝑍, where
𝑋𝑌 is the base. Line segment 𝐴𝑍 is still the
median because point 𝐴 is the midpoint of 𝑋𝑌. But because the line segment 𝐴𝑍
is perpendicular to the base 𝑋𝑌, in this case, 𝐴𝑍 is also the height. This fact where the median and
height are the same value is true in isosceles triangles. But since we haven’t been told
whether or not triangle 𝑋𝑌𝑍 is isosceles, the only thing we can say for sure is
that 𝐴𝑍 is the median. We cannot tell if it is the height
or not, which makes option D median the best answer.
