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Relation between Area of Parallelogram and 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 IMO Subjects

Concept Explanation

Relation between Area of Parallelogram and Triangle

Theorem: If a triangle and a parallelogram are on the same base and between the same parallels the area of the triangle is equal to half of the parallelogram.

Given A $Delta ABC$ and a $parallel ^{gm} BCDE$ on the same base BC and between the same parallels BC and AD

To prove $ar(Delta ABC)=frac{1}{2}ar(parallel ^{gm}BCDE)$

Construction Draw $AL perp BC;;and;;DM perp BC$ , meeting $BC$ in $M$

Proof Since A, E and D are collinear and BC || AD

$therefore$ $AL=DM$ ...(i) [ $because$ Distance between parallel lines is always same]

Now, $ar(Delta ABC)=frac{1}{2}(BCtimes AL)$

$Rightarrow ar(Delta ABC)=frac{1}{2}(BCtimes DM)$ ............(1) $[because AL=DM (from(i))]$

$large ar(parallel ^{gm}BCDE)= BC ;times;DM$ ...............(2)

From Eq (1) and Eq (2)

$Rightarrow ar(Delta ABC)=frac{1}{2}ar(parallel ^{gm}BCDE)$

ILLUSTRATION P and Q are any two points lying on the side DC and AD respectively of a parallelogram ABCD. Show that ar(APB)= ar(BQC)

Solution: Now $large parallel ^{gm}$ ABCD and $large Delta$ ABP lie on the same base AB and between the same parallel AB || CD

$Rightarrow ar(Delta ABP)=frac{1}{2}ar(parallel ^{gm}ABCD)$ .................. (1)

Similarly $large parallel ^{gm}$ ABCD and $large Delta$ BCQ lie on the same base BC and between the same parallel BC || AD

$Rightarrow ar(Delta BCQ)=frac{1}{2}ar(parallel ^{gm}ABCD)$ .................. (2)

From Eq (1) and Eq (2)

$ar(Delta ABP)=ar(Delta BCQ)$

Hence Proved

Sample Questions

(More Questions for each concept available in Login)

Question : 1

A triangle and a parallelogram are on the same base and between the same parallels. The ratio of the areas of triangle and parallelogram is
A. 1 : 1	B. 1 : 2	C. 2 : 1	D. 1 : 3
Right Option : B
View Explanation

Question : 2

In the figure if $large dpi{110} AD left | right |BC$ , AD=BC and CN is perpendicular to AB (produed),which of the following reationalship is true?

$ar(ACB)=frac{1}{2}ar(ABCD)$

ar(ABC)=ar(ABCD)

ar(ABCD)=ar(ANCD)

ar(ADC)=ar(ABCD)

Right Option : A

View Explanation

Question : 3

In the given figure, p || q and BC || AD.

if ar $large dpi{110} fn_phv Delta ABE=16 ;cm^2$ ,

Then ar(ABCD) is equal to ______________

16 $cm^2$

48 $cm^2$

32 $cm^2$

8 $cm^2$

Right Option : C

View Explanation

Video Link - Have a look !!!

Language - English

Chapters

Polynomials II

Direct and Inverse Variation

Linear Equations in One Variable

Introduction to Trigonometry

Introduction to Euclids Geometry

Herons Formula Complete

Polynomial I

Surface Area and Volume II

Surface Area and Volume I

Coordinate Geometry

Constructions

Areas Of Parallelograms And Triangles II

Areas Of Parallelograms And Triangles I

Linear Equations in Two Variables

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