Altitude of the Triangle: Formula and Properties

The altitude of a triangle can be defined as a perpendicular line segment drawn from the vertex to the opposite side of the triangle.

• Based on the triangle types, the altitude's position can be changed. 
• There are three altitudes of a triangle, this altitude can be used to find the height of the triangle.
• It forms the right angle with the base. 

The formula for expressing the altitude of the triangle is 

Altitude = \(\frac{2 \times Area}{ Base}\)

Key Terms: Altitude of triangle, perpendicular, line segment, right angle, Vertex

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What is the Vertex of a Triangle?

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The point of interaction, of any two sides of a triangle is called the vertex of the triangle. 

Vertex of a triangle

The properties of a vertex are: 

• A triangle has three vertexes. 
• A line can be drawn from the opposite side to the vertex of a triangle.
• In an equilateral triangle, the perpendicular line from the middle point from the base can be drawn from any side of the triangle. 
• But in the case of an isosceles triangle, the base from where the perpendicular can be drawn is not equal in length. 

Read more: Similarity of triangles

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What is the Altitude of a Triangle?

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The altitude of a triangle is the line drawn perpendicular to the base from the opposite vertex. The line makes a right angle to the triangle and is called the height of the triangle. 

Altitude of a triangle

The latitude of the triangle can be used to calculate the area of the triangle. The area of the triangle can be given by, ½ base * height. Now, the base can also be calculated with the formula: Base= 2 x area/height

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Properties of Altitude of a Triangle 

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The properties of the altitude are as follows:

• There can be three altitudes of a triangle, in the case of an equilateral triangle all the side are equal so altitude can be calculated from any side.
• The altitude is perpendicular to the opposite side of the triangle. 
• The altitude of the triangle can be both inside and outside the triangle. 
• The triangle's orthocenter is where the three altitudes intersect.

Also Read: Height and Difference

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Altitudes of Triangles 

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The altitudes of different types of triangles are as follows:

 

Obtuse triangle

The obtuse triangle has one of the angles more than 90 degrees. For an obtuse triangle, the altitude of the triangle is outside. For this, we need to extend the triangle's base and then draw a perpendicular line. 

Obtuse triangle

Equilateral Triangle

 

An equilateral triangle has all the sides equal. The latitude of an equilateral triangle bisects the base and also the angle of the vertex. 

Equilateral Triangle

For an equilateral triangle:

In triangle ADB,

sin 60° = h/AB

∴ sin 60° = h/6

√3/2 = h/6

h = (√3/2)6

The height of the triangle will be (√3/2) * side of the triangle. 

Right Triangle 

For an equilateral triangle, the altitude is done on the hypotenuse of the triangle, and is the geometrical mean of the line segment. The segment to the hypotenuse form two similar triangle. This is called as right triangle altitude theorem. 

Right Triangle

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What is the median of a Triangle?

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The median of a triangle is the line segment that joins the vertex of the triangle to the midpoint of the opposite side. 

Median of a Triangle

Also check:

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Things to Remember

• Altitude is the height of the triangle. 
• The altitude of the triangle can be both inside and outside the triangle. 
• In the case of an obtuse triangle, extend the base of the triangle and then draw a perpendicular to find the altitude. 
• Altitude can be given by Altitude = \(\frac{2 \times Area}{ Base}\)
• The altitude can also be used to find the area of the triangle. 
• There can be a maximum of three altitudes of a triangle. 
• For an equilateral triangle, the altitude can be made from any side of the triangle.

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