Question

Find area of rhombus ABCD if the length of its diagonal is 126 cm and perimeter is 260 cm.

Answer 1

Hint: Use formula to find out the area of rhombus$ = \dfrac{1}{2} \times {d_1} \times {d_2}$, where
${d_1} = $Length of 1st diagonal
${d_2} = $Length of 2nd diagonal


Complete step by step solution:

Length of one diagonal $ = 126\,cm$
Perimeter $ = 260\,cm$
Since Perimeter$ = 4 \times side$
$
   \Rightarrow side = \dfrac{{perimeter}}{4} \\
   = \dfrac{{260}}{4} \\
   = 65\,cm \\
 $
We know that diagonals of rhombus bisects each other at $90^\circ .$
So, $DO = OB = \dfrac{{126}}{2} = 63\,cm$
Now,
In $\Delta AOB$, we apply Pythagoras theorem,
\[
  {(AB)^2} = {(AO)^2} + {(BO)^2} \\
   \Rightarrow {(65)^2} = {(AO)^2} + {(63)^2} \\
   \Rightarrow 4225 = {(AO)^2} + 3969 \\
   \Rightarrow {(AO)^2} = 4225 - 3969 \\
   \Rightarrow {(AO)^2} = 256 \\
   \Rightarrow AO = \sqrt {256} \\
   \Rightarrow AO = \sqrt {16 \times 16} \\
   \Rightarrow AO = 16\,cm \\
 \]
Since length of $AC = 2AO$
$
  \therefore AC = 2 \times 16 \\
   = 32\,cm \\
 $
Now area of rhombus
$
   = \dfrac{1}{2} \times {d_1} \times {d_2} \\
   = \dfrac{1}{2} \times 32 \times 126 \\
   = 2016\,c{m^2} \\
 $
$\therefore $Area of rhombus $ = 2016\,c{m^2}$


Note: Students must be aware of the fact that “Every square is a rhombus but not every rhombus is a square”. Opposite angles of rhombus are equal and diagonals bisect each other at \[90\]degrees whereas in case of a square, measure of each angle is equal to \[90\]degrees.