Medians and Altitudes of a Triangle

Here we will discuss about Medians and Altitudes of a Triangle 

Median of a Triangle:

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.

Definition of Median of a Triangle:

A line segment joining a vertex of a triangle to the mid-point of the opposite side is called a median of a triangle.

In the above figure, AD is a median of ∆ABC i.e., BD = DC.

A triangle has three medians.

Solved Example on Median of a Triangle:

A Geometrical Property of Medians of a Triangle:

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 of a Triangle:

Before studying altitude of a triangle, let us first know about the perpendicular lines.

Two lines m and n are said to be perpendicular to each other, if one of the angles measured by them is a right angle. We read as n is perpendicular to m and write n ⊥ m.

Also, let P be the mid-point of AB. Then, we can say that the line n is the perpendicular bisector of the line segment AB.

REMEMBER

Two rays or two line segments are said to be perpendicular to each other, if the corresponding lines or segments determined by them are perpendicular.

For example, the foot of a table is perpendicular to the surface of the table.

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.

Definition of a Altitude of a Triangle:

The perpendicular line drawn from vertex of a triangle to its opposite side is called an altitude.

The side on which the altitude drawn is called base.

In the adjoining figure, AL ⊥ BC. So, BC is the base and AL is the corresponding altitude of ∆ABC.

REMEMBER

There can be one perpendicular from each vertex of a triangle to the opposite side. Thus, there are three altitudes in a triangle.

Let us discuss about three altitudes in a triangle.

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.

Examples on Medians and Altitudes of a Triangle:

1. In the adjoining figure, name the altitude and median of ∆ABC.

Solution:

In the above figure, we find that AD ⊥ BC. Hence, AD is the altitude of AABC.

Also since CE is the bisector of AB i.e. AE = BE.

Therefore, CE is the median of  ∆ABC.

9th Grade Math

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