Question

Sanya has a piece of land which is in the shape of rhombus. She wants her one daughter and one son to work on the land and produce different crops. She divided the land in two equal parts. If the perimeter of the land is 400m and one of the diagonals is 160 m, how much area each of them will get for their crops.

Answer 1

Hint: In this question, we have to find the area of the triangle formed by the diagonals of the rhombus. As we know that all the sides of the rhombus are equal and the area of any triangle with the given sides can be found by using the Heron’s formula:
$\sqrt {s(s - a)(s - b)(s - c)} $ square unit.
[Where S is semi-perimeter and a, b, c are the lengths of the sides.]
We also know that semi-perimeter is
$s = \dfrac{{a + b + c}}{2}$

Complete step-by-step answer:
As the lengths of sides of the rhombus are equal, we can find the length of each side by the given perimeter
To find the length of side of a rhombus, we have to use the formula
i.e., $\dfrac{{perimeter}}{4}$
Therefore, the length of side of the rhombus is
$\dfrac{{400}}{4}$
$ \Rightarrow 100$m
We also know that one diagonal of rhombus divides it in two equal parts and we get a triangle
Therefore, \[a = 100,b = 100\& c = 160\]
i.e., lengths of sides of triangle are \[100m,100m{\text{ }}and{\text{ }}160m\]
Find the semi-perimeter:
To find the area of the triangle with the sides $100$m,$100$m and $160$m, we can use the Heron’s formula
i.e., $\sqrt {s(s - a)(s - b)(s - c)} $ square unit.
\[{
   \Rightarrow \sqrt {180(180 - 160)(180 - 100)(180 - 100} \\
   \Rightarrow \sqrt {180(20)(80)(80)} \\
   \Rightarrow \Delta = 4800 \\
} \][Where $\vartriangle $= Area of the triangle]
Hence, answer is $4800$ ${m^2}$

Note: Whenever we face such types of problems the key concept is to use Heron’s formula as stated in the solution. One can go wrong while calculating the square root. We must take care about the calculations. Alternative, we can also find the side of the rhombus by using area’s formula-
$\dfrac{1}{2} \times $product of diagonals.