Medians and altitudes in triangles are special types of line segments that form additional points and properties within a triangle once they are drawn. The special points formed by medians and altitudes can be extremely useful in real life, especially if you're trying to find the exact center of a surface or object. In fact, these types of problems may require you to create or locate special points within a structure by drawing medians or altitudes.
Every triangle has three sides and angles and therefore three medians and three altitudes. Remember these basic properties when using medians and altitudes to solve a problem.
Medians
Altitudes
Each median connects a vertex with the midpoint of the side opposite the vertex. The median therefore divides the opposite side into two congruent line segments. The intersection of three of medians form a point called the centroid.
In the diagram above, triangle ABC has three medians: BF, CE and AD.
| BP is \(\small\mathsf{ \frac{2}{3} }\) of BF and BP is twice PF. CP is \(\small\mathsf{ \frac{2}{3} }\) of CE and CP is twice PE. AP is \(\small\mathsf{ \frac{2}{3} }\) of AD and AP is twice PD. |
Each altitude connects a vertex with the side opposite the vertex but at a right angle. The intersection of three of altitudes forms a point called an orthocenter.
In the diagram above, triangle BCH has three altitudes BD, CF and HE. Each altitude forms a right angle on the line containing the opposite side.
Take a look at this problem. How can your knowledge of triangle medians help you solve it?
Your neighbor, a father to be, asks for your help in hanging a mobile over his baby's crib. He's tried several times, but he can't get the mobile to hang evenly. How can medians or altitudes help you solve your neighbor's problem before the baby arrives?
Draw the triangle ABC to represent the mobile, and describe where the middle point should be located to perfectly balance the mobile.
The triangle ABC has three medians CF, AD and BE with centroid point G. G is the point from which you should hang the mobile.
