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Constructing Altitudes - Problem 1
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An altitude is a line segment in a triangle from a vertex to the side opposite that vertex, and perpendicular to that side. So, in order to construct an altitude, first swing an arc from the vertex that is large enough to intersect the opposite side twice. Then, from each point at which the arc intersects that side of the triangle, making sure that the arcs at each of these points of intersection are the same size. Connect the point at which these arcs intersect to the vertex of the triangle, and extend the line to the opposite side of the triangle. This creates a line perpendicular to at side of the triangle, creating an altitude.

You’re probably going to be asked to construct an altitude in a triangle. So in construction remember you’re only using two things a compass and a straightedge, but what is an altitude? Well we said our definition of an altitude is a perpendicular segment from a vertex to the line containing the opposite side.

So if I look at this triangle right here, we’re being asked to construct altitude BD, so that tells you the vertex that you’re going to. So we’re starting at B and we’re going to some point D on this opposite side.

So an idea of what we’re going to do here is to think of the vertex B as some point in space and we have this opposite side AC. To do this construction, we would swing an arc from that point, and then from each of these endpoints we would swing two more arcs and then we would connect these and we would have our perpendicular segment, so that’s what we’re going to do.

Start off my grabbing your compass and you want to swing an arc with your compass sharp end at B and we want to intersect this side AC twice. So sharp end is going to be at B and I’m going to extend that a little bit more so I get two points of intersection okay actually it’s a little too much, so I’m going to swing an arc, so now we have our two points of intersection and I need to have, In have one point on this line that’s vertex B and I’m to swing one more arc from each of these end points, so here is one arc.

Come over to this point of intersection swing another arc and so now we have our two points. We have a point down here and we have B and notice that it says line segment BD, so I’m going to make sure that this is going to end on that side AC. So now I can connect these, I’m going to label it as a right angle, and I’m going to create point D.

Key thing here was remembering that constructing an altitude is the exact same process as constructing a perpendicular through a point to a line.

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