Example 2 (Method 1)
If a triangle and a parallelogram are on the same base and between the same parallels, then prove that area of triangle is equal to half the area of parallelogram.
Given: A parallelogram ABCD
and ABP on the same base AB
and between the same parallels
To prove: Area of triangle is equal to half the area of parallelogram.
ar ( ABP ) = 1/2 ar (ABCD)
Construction: Join DP
Let DM AB & PN AB
Proof:
Example 2 (Method 2)
If a triangle and a parallelogram are on the same base and between the same parallels, then prove that area of triangle is equal to half the area of parallelogram.
Given: A parallelogram ABCD
and ABP on the same base AB
and between the same parallels PC & AB
To prove: Area of triangle is equal to half the area of parallelogram.
ar ( ABP ) = 1/2 ar (ABCD)
Proof: In parallelogram ABCD,
AB CD
So, PC AB
We draw a line BQ parallel to AP ,
i.e. , BQ AP
Since BQ AP
& PQ AB,
Both pairs of opposite sides are parallel
ABQP is a parallelogram
Parallelograms ABQP & ABCD are on the same base AB
and between the same parallel lines AB & PC
Area(ABQP) = Area(ABCD)
In parallelogram ABQP,
BP is the diagonal
So, ABP QBP
Area( ABP) = Area( QBP)
Now,
Area( ABP) = Area( QBP) = 1/2 Area(ABQP)
Area( ABP) = 1/2 Area(ABQP)
Area( ABP) = 1/2 Area(ABCD)
Hence proved

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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