Medians and Altitudes of a Triangle

Here we will discuss about Medians and Altitudes of a Triangle 

Median:

The straight line joining a vertex of a triangle to the midpoint of the opposite side is called a median. A triangle has three medians. Here XL, YM and ZN are medians.

Medians of a Triangle


A geometrical property of medians:

The three medians of a triangle are concurrent, i.e., they have a common point of intersection. This point is known as the centroid of the triangle. It divides each median into the ratio 2 : 1.

Here, the three medians intersect at G.

Thus, G is the centroid of the triangle.

Also, XG : GL = 2 : 1

          YG : GM= 2 : 1

and    ZG : GN = 2 : 1



Altitude:

An altitude of a triangle, with respect to (or corresponding to) a side, is the perpendicular line segment drawn to the side from the opposite vertex.

XL is the altitude with respect to the side YZ.

YM is the Altitude

YM is the altitude with respect to the side ZX.

ZN is the Altitude

ZN is the altitude with respect to the side XY.

Altitude of Right-angled Triangle

If ∆XYZ is a right-angled triangle, right angled at Y, XY is the altitude with respect to YZand YZ is the altitude with respect to XY.

Altitude of Obtuse-angled Triangle

If ∆XYZ is an obtuse-angled triangle in which ∠XYZ is the obtuse angle, the altitude with respect to YZ is the line segment XM drawn perpendicular to ZY produced.

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9th Grade Math

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