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How do you derive the area formula for a parallelogram?

Answer
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Hint: At first, we divide the parallelogram into two triangles by joining any two opposite vertices. These two triangles are exactly the same (congruent) and thus have equal areas. The area of the parallelogram is the summation of the individual areas of the two triangles. We drop a perpendicular from a vertex to its opposite side to get an expression for the height of the triangles. The area of the individual triangle is 12×base×height .The area of the parallelogram being twice the area of the triangle, thus becomes after evaluation base×height .
Complete step by step answer:
The parallelogram can be divided into two triangles by constructing a diagonal by joining any two opposite vertices.
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In the above figure, ΔABD and ΔBCDare the two such triangles. These two triangles have:
AB=CD (as opposite sides of a parallelogram are equal)
AD=BC (opposite sides of a parallelogram are equal)
BD is common
Thus, the two triangles are congruent to each other by SSS axiom of congruence. Since, the areas of two congruent triangles are equal,
area(ΔABD)=area(ΔBCD)
Now, we need to find the area of ΔABD . We draw a perpendicular from D to the side AB and name it as DE . Thus, ΔABD is now a triangle with base AB and height DE .
Then, the area of the ΔABD becomes
area(ΔABD)=12×base×height
area(ΔABD)=12×AB×DE
The area of ΔBCD being equal to that of ΔABD is also 12×AB×DE .
The area of the parallelogram ABCD is the summation of the areas of the two triangles, which is
area(||gmABCD)=2×12×AB×DE
area(||gmABCD)=AB×DE
It is to be noted that the perpendicular DE is also the height of the parallelogram ABCD and the base AB is also the base of the parallelogram ABCD .
Thus, we can write the general expression of the formula of the area of a parallelogram as
area(||gmABCD)=base×height
Where base indicates any one of the sides and height indicates the perpendicular distance between this side and its opposite side.

Note: Students must be careful while drawing the perpendicular and must remember that perpendicular must be drawn from the opposite side to the side taken as base. Area of a parallelogram can also be derived by assuming any two adjacent sides of it as two vectors say a and b . Then, the area becomes
area(||gmABCD)=a×b
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