Altitude of a Triangle

A triangle's altitude is a perpendicular traced from the triangle's vertex to the opposite side. Because a triangle has three sides, three altitudes may be drawn in it. The elevations of various triangles differ from each other. The altitude of a triangle, also known as its height, is used to calculate the area of a triangle and is represented by the lowercase letter 'h'.

The height of a triangle is the orthogonal line segment extended from the triangle's tip to the opposing side. The altitude forms a right angle with the bottom of the triangle on which it rests. It is calculated by measuring the distance between the vertex and its opposite side. It should be noticed that three elevations may be derived from each of the triangle's vertices.

Note that an altitude is completely different from a median in numerous ways.

• A triangle's median is the line segment traced from its vertex to its opposing side, whereas its height is the perpendicular distance from its base to its opposite vertex.
• A median splits a triangle into two halves, but an altitude does not.
• The place where the three medians of a triangle join are defined as the triangle's centroid, whereas the position where the three heights of the triangle coincide is regarded as the triangle's orthocenter.
• The median is always within the triangle; however, the altitude may be outside or inside, depending on the type of triangle.

Now that we have studied various concepts of altitude and the major differences between the height of a triangle and median let us now study some important properties about it. They are as follows:

• All the triangles can have a maximum of three altitudes.
• The perpendicular height may lie on the interior or exterior of a triangle based on the type of the triangle.
• The altitude is perpendicular and hence makes a right angle with the base it falls on.

Before learning the different formulas related to altitude, let us study some rudimentary concepts from which all the other formulas are derived. We are aware that the area of a triangle is equal to half of its base and height, which is expressed as A = ½ x b x h; from this, we can concur that h = (2 x A)/ base. This is helpful in deriving other formulas. Let us see the formulas related to isosceles, equilateral, scalene, right-angle, and obtuse-angled triangles.

Altitudes of Different Types of Triangles

Scalene Triangle: Since a scalene triangle has different values of sides, we calculate the height of a scalene triangle using Heron's formula to find the area and then substitute the value of the area in the formula mentioned above to get h = 2√s(s−a) (s−b) (s−c)/b.

Isosceles Triangle: The height of an isosceles triangle having sides' a' and 'b' is determined using the Pythagoras theorem h =√a2−b2/4.

Equilateral Triangle: In an equilateral triangle, the height is determined using the formula h = (a√3)/2

Right-angled Triangle: The right-angled triangle's height is formulated as h = √xy.

Obtuse Triangle: The altitude of an obtuse angle triangle lies on the exterior from the area of the triangle. The height is calculated using the basic formula of the area and base of the triangle.