Construct Triangle Segments: Altitude, Median, Angle Bisector MathBitsNotebook.com Topical Outline | Geometry Outline | MathBits' Teacher Resources Terms of Use   Contact Person: Donna Roberts

Remember -- use your compass and straightedge only!

You already know how to make these constructions. Just remember the definitions of these terms, and you will have a hint as to how to create their constructions.

By definition, an altitude of a triangle is a segment from any vertex perpendicular to the line containing the opposite side.

The definition tells us that the construction will be a perpendicular from a point off the line. The "point" will be the vertex of the triangle from which the altitude will be drawn, and the "line" will be the side of the triangle to which the altitude will be perpendicular.

Altitude in an Acute Triangle Construct an altitude from vertex C. Notice that it was necessary to extend the side of the triangle from A through B to intersect with our arc. The altitude is a segment (not a line), so be sure to label the portion of the construction that is actually the altitude, or refer to it by name, .

Altitude in an Obtuse Triangle Construct an altitude from vertex E. Notice that it was necessary to extend the side of the triangle from F through G to intersect with our arc. Remember, in an obtuse triangle, your altitude may be outside of the triangle. Be sure to label the altitude, such as , or indicate that it is a segment.

By definition, a median of a triangle is a segment joining any vertex of the triangle to the midpoint of the opposite side.

The definition contains the word "midpoint" which tells us that the opposite side will be bisected. We will construct the bisector of the opposite side in the triangle, to locate the midpoint, and then simply connect the midpoint to the vertex to create the median.

Median in an Acute Triangle Construct a median from vertex C. Sinceis the side opposite vertex C, we bisected to locate its midpoint M. Now, simply connect midpoint, M, to vertex C.

Median in an Obtuse Triangle Construct a median from vertex E. Since is the side opposite vertex E, we bisected to locate its midpoint, M. Now, simply connect midpoint M to vertex E to create the median of the triangle. Remember that the medians of an obtuse triangle remain inside the triangle (unlike its altitudes).

By definition, an angle bisector is a ray from the vertex of an angle, into the interior of the angle, forming two congruent angles. Since a triangle contains 3 angles, it will have 3 angle bisectors.

This construction will be a direct application of the construction "bisect an angle".

Angle Bisector in an Acute Triangle Construct an angle bisector of ∠C. When constructing the angle bisector in a triangle, it is customary to refer to only the portion of the bisector which lies inside the triangle, such as . Note: an angle bisector does not necessarily bisect the side of the triangle. We do not know if AE = EB. We only know that ∠ACE ∠BCE.

Angle Bisector in an Obtuse Triangle Construct an angle bisector of ∠E. The angle bisector will be . We know that ∠GEH ∠FEH. We do not know if H bisects the side from G to F. An angle bisector in a triangle is not necessarily also a median for that triangle.