Altitude and Median of Triangle: Definition & Properties

Altitude and median of a triangle are two distinct concepts that are often confused to be equal or the same. Though both of them have equal importance while studying different aspects of triangles, they are not the same rather than in a rare case. Median begins from a vertex whereas an altitude is often referred to as beginning from a side of the triangle. But both the median and altitude touch the vertex and a side of the triangle. Some of the factors make median and altitude interesting parts of a triangle. 

Key Terms: Triangle, Perpendicular Line, Heights, Median, Altitude, Area of Triangle

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Triangle

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Triangle is a closed geometric figure with three sides. The points at which the sides meet are known as the vertex (plural, vertices). A triangle may form any angle at the vertices, but the sum of all angles will be equal to 180°. With the help of the length of a side and altitude, one can measure the area of a triangle.

Also read: Triangle and its types

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Altitude of a triangle

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Altitude is the perpendicular line drawn from a vertex to the opposite side of the triangle. A perpendicular line will always make a 90â° with the side on which it intersects. Since a triangle has three sides and three vertices, a triangle would have 3 altitudes. Altitude helps in calculating the area of a triangle,

 Area of a triangle = ½ x Base x Altitude

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Median of a triangle

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The median of a triangle is a line segment drawn from a vertex to another point on the opposite side of that vertex so that the line segment divides the opposite side into two halves.

In the above figure, AR is a line segment that divides side BC into two halves, that is, BR and RC. Similarly, all the three medians divide the sides into equal halves.

With the help of the length of sides, we can find the length of a median, using the below formula:

Length of median = √(2b²+ 2c²- a²)/4

Where, a, b and c are sides of the triangle and ‘a’ is the side to which the median is drawn.

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Properties of median and altitude

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The altitude and median of a triangle have different properties as a part of the triangle. The properties can be different for different kinds of triangles, but there are some features or key properties that make them identifiable.

• Like the median, there would be 3 altitudes in a triangle. In other words, the number of sides in a triangle would be equal to the number of altitudes.
• The angle that an altitude makes with the opposite side would always be 90 degrees.
• If all the altitudes of a triangle are drawn at a time, then the three altitudes will surely intersect at a point called the orthocentre.
• Altitude is a part of a triangle that always needn’t be within the sides of a triangle.
• Altitude is a basic component that helps to calculate the area of a triangle.
• In an equilateral triangle, the altitude is the same as the median.

There are some properties of a median that may be defined as its characteristics. These properties include:

• An altitude begins from the vertex of a triangle and ends at the point of the opposite side which divides the opposite side equally.
• The area of the two parts formed by drawing a median (which would be two triangles) would always be equal.
• The number of medians in a triangle would be always equal to the number of vertices in the triangle.
• If three medians are drawn at a time, they will surely meet at a point of the triangle.(Centroid) 
• Areas of the number of smaller portions that a median creates within a triangle would be always equal.
• In an equilateral triangle, the median is the same as the altitude.

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 Things to remember

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• The altitude of a triangle is the perpendicular line drawn from a vertex of the triangle to the opposite side. In other words, altitude represents the height of the triangle.
• Altitude is used to calculate the area of a triangle, together with the length of the base to which the altitude is drawn.
• A Median is a line segment drawn from the centre of a side of the triangle to the opposite vertex. In other words, the median divides the side into two halves.

Length of a median can be calculated by the formula,

Length of median = √ (2b²+ 2c²- a²)/4

Where, a, b and c are sides of the triangle and ‘a’ is the side to which the median is drawn.

• In the case of an equilateral triangle, the median of the triangle is the same as the altitude of the triangle.

Also read: Equilateral triangle

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