Altitude of a Triangle

The altitude of a triangle is the perpendicular line segment that is drawn from the vertex of a triangle to the opposite side known as the base, or the line containing the base. Note that the altitude may be perpendicular to the base, or to the extension of the base.

As you will see in the illustrations below, the altitude of a triangle can be found in three (3) different locations or places. It may lie inside, outside, or be a part of the triangle itself as in the legs of a right triangle.

Case 1: Altitude of Triangle Found Inside

triangle ABC has an altitude that is inside

Line segment \overline {BO} is an altitude formed inside triangle ABC. Notice that \overline {BO} makes a right angle with the opposite side, line segment \overline {AC}

Case 2: Altitude of Triangle Found Outside

triangle KLM has altitude that is found outside

It is possible for an altitude to lie outside the triangle. In this case, line segment \overline {KO} is an altitude formed outside triangle KLM. Observe that altitude \overline {KO} makes a right angle with the extension of the base \overline {LM}.

Case 3: Altitude is Part of the Triangle

triangle XYZ has an altitude that is part of the triangle itself

In the last two cases, the altitude is either formed inside or outside the triangle. However, it is possible for the altitude to be part of the triangle itself. This happens if the triangle involved is a right triangle. A right triangle is a triangle such that one of its internal angles has a measure of 90 degrees. The reason is that the legs of a right triangle are naturally perpendicular to each other.

Notice that line segment \overline {XY} is a side of triangle XYZ which serves as the altitude as well.

In fact, if we rotate the triangle such that \overline {ZY} is the altitude, it follows that segment \overline {XY} becomes the base.

Orthocenter of an Acute Triangle

For an interesting side note, if we draw all the altitudes of an acute triangle, they will intersect at a common point inside the triangle. This point of intersection is known as the orthocenter.

orthocenter is the point of intersection of three altitudes

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Area of a Triangle