Altitude vs Median
Altitude and median are two heights used when discussing the geometry of a triangle.
Altitudes of a Triangle
Altitude of a triangle is a line segment perpendicular to a side and passing through the vertex opposing the side. Since a triangle has 3 sides, they each have a unique altitude per side giving a total of 3 altitudes per triangles. The side to which the altitude is perpendicular is known as the extended base of the altitude.
Altitude is commonly denoted by the letter h (as in height).
Altitudes are specifically used in calculating the area of triangles. The area of a triangle is half the product of the altitude and its base.
Area = 1/2 altitude×base = 1/2 h×b
Also, the point of intersection of the three altitudes from the sides is known as the orthocenter. The orthocenter lies within the triangle if and only if the triangle is an acute triangle.
Medians of a Triangle
A median is a line segment passing through the midpoint of a side and the vertex opposing that side. The median bisects the angle of the vertex. It also divides the area of the triangle in half. Likewise the altitudes, there is a unique median for each side; therefore every triangle has three medians. All the three medians together divide the triangle into six smaller triangles with the same area. (Refer diagram)
The three medians of the triangle intersect at a point, which divides each median to 2:1 ratio. It is known as the centroid of the triangle and, for a uniform laminar triangle the center of mass is located here.
Both the orthocenter and the median lie on the Euler line, which also contains the circumcenter of the triangle.
What is the difference between Altitude and Median?
• Both altitude and median pass through a vertex, but altitude pass through the opposing side at right angles; i.e. perpendicular to the side, while median pass through the midpoint of the opposing side.
• Altitude is used to calculate the area of the triangle.
• A single median divides the area of the triangle in half and all three divides the triangle into six smaller triangles with equal area.
• The medians intersect at the centroid, while altitudes intersect at the orthocenter.
• The orthocenter may lie inside or outside the area of the triangle, but centroid always lies within the area of the triangle.