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Maths / Areas Of Parallelograms And Triangles / Relation between Area of Parallelogram and Triangle

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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

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Relation between Area of Parallelogram and Triangle

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