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Median and Altitude of Triangle

Areas Of Parallelograms And Triangles II - Concepts

Median and Altitude of Triangle

Triangles on Same Base and Between Same Parallels

Relation between Area of Parallelogram and Triangle

Class - 9th CBSE Subjects

Concept Explanation

Median and Altitude of Triangle

Median and altitude of triangle: A median is a line segment drawn from the vertex of a triangle to the midpoint of the opposite side of triangle. As in a triangle there are three vertices and three sides, therefore there are three medians in each triangle. From each vertex we can draw a line joining the midpoint of the opposite side of the triangle. It divides the opposite side into two equal line segments. In the figure AE is a median as E is the midpoint of the side BC. Thus BE = EC

Altitude: An altitude is a perpendicular line segment drawn from a vertex of a triangle to the opposite side. There are three altitudes in a triangle as there are three vertices and three side. In the figure AD is the altitude from vertex A. AD is perpendicular to BC.

Theorem: Show that median of a triangle divides it into two triangles of equal area.

Given: A triangle ABC with AE as the median

To Prove: ar ( $large Delta$ ABE) = ar ( $large Delta$ ABE)

Construction: Draw AD $large perp$ BC

Proof: $large Area ;of; a; triangle; =frac{1}{2};X ;base;X; height$

$large ar (Delta ABE); =frac{1}{2};X ;BE;X; AD$ ..........(1)

$large ar (Delta AEC); =frac{1}{2};X ;EC;X; AD$ ........(2)

As AE is a median $large Rightarrow$ BE = EC

From Eq (1) and (2) ar ( $large Delta$ ABE) = ar ( $large Delta$ ABE)

Hence Proved.

ILLUSTRATION: In the figure E is the mid point of the median AD of triangle ABC. Prove that:

$large ar(Delta BDE)=frac{1}{4} ar (Delta ABC)$

Solution: It is given that AD is the median

we know that the median divides the triangle into two triangles of equal area

$large therefore ar(Delta ABD)=frac{1}{2} ar (Delta ABC)$ .....(1)

Also E is the midpoint of AD. Therefore AE is the median of triangle ABD

$large therefore ar(Delta BDE)=frac{1}{2} ar (Delta ABD)$

Using equation 1

$large Rightarrow ar(Delta BDE)=frac{1}{2} ;X; frac{1}{2}ar (Delta ABC)$

$large Rightarrow ar(Delta BDE)=frac{1}{4}ar (Delta ABC)$

Sample Questions

(More Questions for each concept available in Login)

Question : 1

If ABC is a triangle and AD is its median lying on the side BC, then
A. ar (ABD = ar (ABC)	B. ar (ABD) = ar (ADC)	C. ar ( ABD) = ar (CAB)	D. ar ( DAC) = ar (ABC)
Right Option : B
View Explanation

Question : 2

For the given triangle ABC, G is the centroid and BC = 60 units. Determine the length of BD.
A. 30	B. 20	C. 40	D. 33
Right Option : A
View Explanation

Question : 3

The median of a triangle divides it into two ________________
A. congruent triangle		B. isosceles triangles
C. right triangles		D. triangles of equal areas
Right Option : D
View Explanation