The “altitude” of a triangle refers to the perpendicular line segment drawn from a vertex of the triangle to the side opposite that vertex. In other words, it is the line segment that connects a vertex to the base (or the side) of the triangle at a right angle.
Key points about the altitude of a triangle:
- Perpendicular: The altitude is always drawn perpendicular to the base or the side of the triangle from the vertex. This means it forms a 90-degree angle with the base.
- Three Altitudes: A triangle has three altitudes, each originating from one of its three vertices. These altitudes may have different lengths depending on the type of triangle (e.g., equilateral, isosceles, scalene) and the specific vertex from which they are drawn.
- Orthocenter: The point at which all three altitudes intersect inside the triangle is called the orthocenter. The orthocenter is not necessarily located inside the triangle for all types of triangles. In an acute triangle (all angles are less than 90 degrees), the orthocenter is inside the triangle. In an obtuse triangle (one angle is greater than 90 degrees), the orthocenter is outside the triangle. In a right triangle (one angle is exactly 90 degrees), the orthocenter is located at the vertex with the right angle.
- Use in Geometry: Altitudes play a crucial role in various geometric calculations and proofs. They are used to find the area of a triangle, determine congruence and similarity of triangles, and solve various geometric problems involving triangles.
In summary, the altitude of a triangle is a line segment drawn from a vertex of the triangle to the opposite side, forming a right angle. It is an important concept in geometry and is used in various geometric calculations and proofs involving triangles.