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# In the following figure, ABCD is a parallelogram and EFCD is a rectangle. Also, $AL \bot DC$. Prove thatA) ${\rm{ar}}\left( {ABCD} \right) = {\rm{ar}}\left( {EFCD} \right)$ B) ${\rm{ar}}\left( {ABCD} \right) = DC \times AL$

Last updated date: 20th Jun 2024
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Hint:
We will first find the area of the parallelogram using the formula for the area of a parallelogram. Then we will find the area of the rectangle using the formula for the area of a rectangle. We will compare both the areas to prove that both the areas are equal.

Formulas used: We will use the following formulas:
1) The area of a rectangle ${A_r} = l \times b$, where $l$ is the length and $b$ is the base.
2) The area of a parallelogram, ${A_p} = b \times h$ , where $h$ is the height and $b$ is the base.

Complete step by step solution:
We will first prove the ${\rm{ar}}\left( {ABCD} \right) = DC \times AL$.
(ii) We will find the area of parallelogram ABCD. We know that the area of a parallelogram is the product of the length of its height and base. So, we can write
$ar\left( {ABCD} \right) = b \times h$
The height of a parallelogram is the perpendicular distance between its 2 parallel opposite sides and the base of the parallelogram is its lowermost side. We can see from the figure that the height of the parallelogram is $AL$ and the base is $DC$. We will substitute these values in the formula for the area of a parallelogram:
$\Rightarrow ar\left( {ABCD} \right) = AL \times DC$
Hence, we have proved ${\rm{ar}}\left( {ABCD} \right) = DC \times AL$.
Now, we will prove the ${\rm{ar}}\left( {ABCD} \right) = {\rm{ar}}\left( {EFCD} \right)$.
We will find the area of the rectangle EFCD. We know that the area of a rectangle is its length times breadth.
$\Rightarrow ar\left( {EFCD} \right) = {\rm{length}} \times {\rm{breadth}}$
Usually, the longer side of a rectangle is taken as its length and the shorter side is taken as its breadth. The breadth of a rectangle is always perpendicular to its length just like the height of a parallelogram. We can see from the figure that the length of the rectangle is $EF$ and the breadth is $ED$. We will substitute these values in the formula for the area of a rectangle.
Therefore,
$\Rightarrow ar\left( {EFCD} \right) = EF \times ED$
We know that the opposite sides of a rectangle are equal in length. So,
$EF = DC$
We can see from the figure that the breadth of the rectangle and the height of the parallelogram are equal. So, we can write
$ED = AL$
So, the area of rectangle EFCD can be rewritten as:
$\Rightarrow ar\left( {EFCD} \right) = DC \times AL$
We know that the area of parallelogram ABCD is also $AL \times DC$.

$\therefore ar\left( ABCD \right)=ar\left( EFCD \right)$

Note:
A parallelogram is a quadrilateral whose opposite sides are parallel and equal. A rectangle is a special kind of parallelogram with all its angles as right angles. The area of a rectangle is also the same as the area of a parallelogram as the base of the parallelogram is the rectangle’s length and the height of the parallelogram is the rectangle’s breadth.