Triangle Altitude Theorem

Altitude of a triangle tutorial here explains the methods to calculate the altitude for the right, equilateral, isosceles and scalene triangle in a simple and easy way to understand.

Altitude of a Triangle Tutorial

Definition

An Altitude of a Triangle is defined as the line drawn from a vertex perpendicular to the opposite side - AHa, BHb and CHc in the below figure. Vertex is a point of a triangle where two line segments meet. For example, the points A, B and C in the below figure.

altitude-of-triangles
Triangles Altitude

In this tutorial, let's see how to calculate the altitude mainly using Pythagoras' theorem. Before that, let us understand the basics of the different types of triangle.

Definition of Equilateral Triangle

An Equilateral Triangle can be defined as the one in which all the three sides and the three angles are always equal.

Definition of Isosceles Triangle

An Isosceles Triangle can be defined as the one in which two sides (AB and AC) are equal in length and the base (BC) is of different length.

Definition of Scalene Triangle

A Scalene Triangle can be defined as the one in which all the three sides are of different lengths.

Definition of Right or Right-angled Triangle

A Right-angled Triangle is the one in which one of the interior angle is right angle (90 degree) and the side opposite the right angle is called the Hypotenuse which is the longest side.

In the below example, ∠ BCA is right angle and BA is hypotenuse.

Calculation

Now, let us calculate the altitude of the right triangle using Pythagoras' theorem. It states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides,

i.e., c2 = a2 + b2.

Right triangle

In the above right triangle, BC is the altitude (height). That can be calculated using the mentioned formula if the lengths of the other two sides are known. Let AB be 5 cm and AC be 3 cm. Let us find the height (BC).

c2 = a2 + b2 52 = a2 + 32 a2 = 25 - 9 a2 = 16 a = 4

Therefore, the height (BC) is 4 cm.

Similarly, the altitude can be calculated for an equilateral using the same method.

In the below example, let the length of the sides be 4 cm. Draw the altitude from B forming BD. The altitudes drawn in an equilateral and isosceles triangles touch the midpoint of the opposite side dividing the side into two equal halves and form two right-angled triangles.

Equivalent triangle

Let us take BDC to find the height (BD) with DC being 2 cm (half of AC). BC2 = BD2 + DC2 42 = BD2 + 22 BD2 = 16 - 4 BD2 = 12 BD = 3.46

Hence the height of the above equilateral is 3.46 cm.

Similarly, the height can be calculated for the Isosceles using the same method.

However for Scalene triangle, we cannot use the Pythagoras theorem as the altitude will not touch the midpoint of the opposite side and hence we cannot determine the length of the base side of the triangle which we consider for calculation of the altitude.

For the Scalene triangle, the height can be calculated using the below formula if the lengths of all the three sides are given.

h = 2*Area/base.

Whereas the area can be calculated using the formula

formula

with a, b, c being the sides and s being (a+b+c)/2.

This tutorial helps you to understand the different types of triangles and to calculate the altitude.

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