Altitude and Median of a Triangle
Altitude and Median of a Triangle
Altitude and Median of a Triangle:
It divides each median into the ratio 2 : 1. Thus, G is the centroid of the triangle. 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.
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.
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 with respect to the side ZX.
ZN is the altitude with respect to the side XY.
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.
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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