Altitude of a Triangle

Altitude Formula Geometry

A line segment drawn from the vertex of a triangle on the opposite side of a triangle which is perpendicular to it is said to be the altitude of a triangle. An altitude makes a right angle(900) with the side of a triangle. An altitude is also said to be the height of the triangle. images will be uploaded soon.


From the figure, In  ABC a line segment BR drawn from the vertex B to the side AC such that it is perpendicular to AC. This line segment BR is called the altitude of the ABC

The altitude of a triangle is used to calculate the area of a triangle. Every triangle has three altitudes.


Using the altitude of a triangle formula we can calculate the height of a triangle.


Altitude of a Triangle Formula can be expressed as:

Altitude(h)  = Area x 2 / base

 

Where Area is the area of a triangle and base is the base of a triangle. And h is the altitude to be found.

(image will be uploaded soon).


According to different measures of different triangles, there are different types of altitudes of a triangle.


Altitude of an Obtuse Triangle

An obtuse triangle is a triangle having measures greater than 900, hence its altitude is outside the triangle. So we have to extend the base of the triangle and draw a perpendicular from the vertex on the base. The below figure shows the altitude ‘h’ of the Obtuse triangle. (image will be updated soon).


Altitude of Equilateral Triangle

A line segment is drawn from the vertex to the opposite side of a triangle such that it is perpendicular to the side and bisects the side in two equal parts then it is said to be the altitude of an equilateral triangle. From the figure side, BS is the altitude of  equilateral ABC, and AD = DC (image will be updated soon).


Altitude of a Right Triangle

The altitude of a right triangle divides the right-angled triangle into two similar triangles. According to the right triangle altitude theorem, the altitude drawn from the vertex on the hypotenuse is equal to the geometric mean of line segments formed by altitude on hypotenuse. image will be updated soon.


Altitude of an Isosceles Triangle

In the isosceles triangle which have two of its sides congruent its altitude bisects the angle of the vertex and bisects the base. image will be updated soon.


Altitude of a Triangle Formula

Triangle Type

Altitude(h) Formula

Altitude of Equilateral Triangle

h = (½) × √3 × side

Altitude of Isosceles Triangle

h =√(a2−b2/4)

Altitude of a Right Triangle

h =√(xy)

Altitude of an Equilateral Triangle Formula

An equilateral triangle has all its angles = 60°. image will be updated soon.


In triangle ADB,

sin 60° = altitude (h)/AB

We know, AB = BC = AC = s ( from the figure it is given)

∴ sin 60° = h/s 

√3/2 = h/s …….(since sin 600 = √3/2)

h = (√3/2)s

Hence,  Altitude of an equilateral triangle formula= h = √(3⁄2) × s

(Solved examples will be updated soon)


Quiz Time:

  1. Find the altitude for the equilateral triangle when its equal sides are given as 10cm.

  2. Find the altitude of a triangle if its area is 120sqcm and base is 6 cm.

FAQ (Frequently Asked Questions)

1. What is the Difference Between Altitude and Median?

Answer: Altitude and the median are two different things.

Altitude is a line segment drawn from the vertex to the opposite side of a triangle such that it is perpendicular to it, whereas the median is just a line drawn from the vertex of a triangle to the midpoint of the opposite side of the triangle.

A median need not be perpendicular to the side of the triangle. While an altitude need not touch the midpoint.

Intersection of the median is a centroid, while altitudes intersect at the orthocenter.

But in case of some triangles like equilateral triangle the median and altitude are the same.

From the below figure it is clear that altitude and median are two different things.

(Image will be updated soon).

2.What makes an Isosceles Triangle?

Answer: A triangle whose two sides are equal is said to be an isosceles triangle. The base angles of an equilateral triangle are equal to each other.The two equal sides of an isosceles triangle are called ‘legs’ whereas the third unequal side is known as the ‘base’.

(Image will be updated soon).


The above figure is an example of an isosceles triangle, where the equal sides are of length ‘b’ and the unequal side has length ‘a’. And h is the height(altitude) of a triangle.

The altitude of the isosceles triangle can be calculated by the formula

h =√(a2−b2/4)

3. What is an Example of Altitude?

Answer: The distance above the sea level, is a real-life example of altitude. The highest altitude point on the earth is Mount Everest. High-altitude zones are always much colder than the areas near the  sea level. This is because the area with high altitudes has decreased air pressure. The human body reacts to this because less oxygen is available for breathing. People living in high altitudes often have a high risk of altitude sickness. It may have serious consequences as brain or lung damage.