Video: Understanding the Medians of a Triangle

In β–³π‘‹π‘Œπ‘, where 𝐴 is the midpoint of line π‘‹π‘Œ, what name is given to line 𝐴𝑍? [A] base [B] height [C] hypotenuse [D] median


Video Transcript

In triangle π‘‹π‘Œπ‘, where 𝐴 is the midpoint of line π‘‹π‘Œ, 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 π‘Œ and then 𝑍. And this is our triangle π‘‹π‘Œπ‘. If 𝐴 is the midpoint of line π‘‹π‘Œ, then 𝐴 is halfway between 𝑋 and π‘Œ. This also means that the segments π‘Œπ΄ and 𝐴𝑋 are equal in length because the midpoint divides the line π‘‹π‘Œ 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 π‘‹π‘Œ 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 π‘‹π‘Œπ‘ is a right triangle. Even if triangle π‘‹π‘Œπ‘ was a right triangle and 𝐴 was halfway between 𝑋 and π‘Œ, 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 π‘‹π‘Œ. 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 π‘‹π‘Œπ‘, where π‘‹π‘Œ is the base. Line segment 𝐴𝑍 is still the median because point 𝐴 is the midpoint of π‘‹π‘Œ. But because the line segment 𝐴𝑍 is perpendicular to the base π‘‹π‘Œ, 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 π‘‹π‘Œπ‘ 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.

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