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Constructing Altitudes - Problem 2

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Teacher/Instructor Brian McCall
Brian McCall

Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

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.

For a triangle with an obtuse angle, the altitude may be outside of the triangle. In this case, extend the side opposite the angle, and do the same process.

In this problem we’re being asked to construct altitude BD, but what is an altitude? Well an altitude is a perpendicular segment from a vertex to the line containing the opposite side. Mr. McCall why don’t you just write that’s a perpendicular segment to the opposite side? Well the reason why is because when you have a problem like this where you have an obtuse triangle, you’re going to have to extend one side.

Since we’re starting at vertex B and we’re going perpendicular to AC, it’s going to fall somewhere in here. So what I’m going to do is I’m going to extend that side, and the reason why we have to do that is because otherwise we won’t have any room to see where our arc intersects AC.

So let’s start off by taking your compass and you’re going to swing an arc from point B so that it intersects this line right here. So I’m going to start off and I’m going to put a sharp end on point B and I’m going to swing my arc. Now the key thing is to make sure that both of these points are far apart. If they’re really close together, your construction won’t look too good.

So now we need to swing two more arcs from these two points compass stays the same for both of these so I’m going to swing one arc right there I’m going to swing another arc right there so we have our point of intersection and we have our vertex B, so that’s enough information to draw a line. And since it says line segment BD, I want to make sure that I’m drawing a line segment, I’m going to drop this down. I’m going to show that this is a 90 degree angle or a perpendicular segment and last I’m going to label that as point D. So remember in obtuse triangles, some of you altitudes will be outside of your triangle and that’s okay.

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