Math's: Concurrent Lines: Altitudes of a Triangle, Medians of a Triangle

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Altitudes of a Triangle

Perpendicular drawn from a vertex of a triangle to the opposite side is called its altitude. Since there are three vertices in a triangle, we can draw three of its altitudes. are the three altitudes of given below which intersect each other at a point H.

  • The three altitudes of a triangle are concurrent i.e. they pass through the same point.

  • The point of concurrency is called the ‘Orthocenter’ of the triangle

Attitudes of a Triangle

Attitudes of a Triangle

It is also to be noted that the orthocenter will be:

  • in the interior of the triangle for an acute triangle

  • in the exterior of the triangle for an obtuse triangle

  • at the vertex containing the right angle for a right triangle

Medians of a Triangle

A line joining a vertex to the mid-point of the opposite side of a triangle is called its median. In total, three medians can be drawn in a triangle. are the medians in given below which intersect each other at a point .

  • All medians of a triangle are concurrent i.e. they pass through the same point.

  • The point of concurrency divides each of the medians in the ratio 2: 1.

  • The point of concurrency G is called the ‘centroid’ of the triangle

.

Medians of a Triangle

Medians of a Triangle

Problem 12.2.1. ABC is an isosceles triangle such that cm and base cm. If is the centroid of , find

Medians of a Triangle

Medians of a Triangle

Solution: In an isosceles triangle, the median to the base is also the perpendicular to the base.

Therefore, in the given , so that is a right-angled triangle.

In ,

is median to

By Pythagoras Theorem,

cm

(centroid divides the median in the ratio )

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