# Video: Understanding the Medians of a Triangle

In △𝑋𝑌𝑍, where 𝐴 is the midpoint of line 𝑋𝑌, what name is given to line 𝐴𝑍? [A] base [B] height [C] hypotenuse [D] median

03:54

### 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.