# Constructing a Median - Problem 1

A **median** of a triangle is a line segment from a vertex to the midpoint of the opposite side. This can be done by first constructing a perpendicular bisector on the side of the triangle opposite the desired vertex, and marking the point at which the bisector intersects the side of the triangle. Recall that a bisector divides a segment into two equal pieces, so the point at which the bisector intersects a segment is the midpoint of that segment. To draw the median, connect the vertex with this midpoint.

In this problem we’re being asked to construct a median from vertex C to the mid-point of the opposite side and we’re going to label that point D. So remember construction means we can only use a compass and a straightedge. So to give an idea of what we’re going to do here, let’s just draw a little sketch.

So this is the sketch of my obtuse triangle, but what is a median? Well a median is the line segment from the vertex to the midpoint of the opposite side. So if I look at this triangle, the first thing I’m going to have to do is construct the midpoint. That’s going to be the hardest part and then I’ll just connect that point with my vertex C.

So let’s grab our compass and the first thing I’m going to do is bisect that side BA. So the key thing here is making sure that your compass is set more than half the length of BA, so I’m going to swing an arc, actually I’m going to erase a little sketch triangle here just so I have enough room. I’m going to swing an arc from vertex B. I’m going to do the same thing not changing my compass. Swing another arc from vertex A and I have my two points of intersection.

Now I don’t need to draw a line here, this line would be the perpendicular bisector of BA. All I need to do is to find the midpoint. So I’m just going to make a point right there which I know is point D and is the midpoint of that line segment. The last step to construct a median is to connect your vertex and I know I’m starting at C with the midpoint of the opposite side. So the key thing here bisecting the opposite side and then just connecting that with the opposite side.

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