Question

The perimeter of a rhombus is 100 m and one of the diagonals is 40 m then the area of the rhombus in square m
A. 625 sq. m
B. 1000 sq. m
C. 400 sq. m
D. 600 sq. m

Answer 1

Hint: In plane Euclidean geometry, a rhombus is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length.

Complete Step-by-Step solution:
The area of a rhombus is given by the formula as follows
\[=\dfrac{1}{2}\times \left( product\ of\ diagonals \right)\]
Now, another important fact about rhombus is that the diagonals of a rhombus bisect each other at right angles.

As mentioned in the question, we have to find the area of this rhombus.
Now, if we draw the diagonals of this rhombus, then we can see that there is a right angled triangle that is formed inside the rhombus.
Now, the side length of a rhombus can be calculated as follows
\[\begin{align}
  & =\dfrac{100}{4} \\
 & =25\ m \\
\end{align}\]
(Because all the sides of a rhombus are equal in length)
So, the triangle that is formed inside has hypotenuse as 25 m and as the diagonals of a rhombus bisect each other, so one of the sides of this triangle is equal to 20 m.
Now, using the Pythagoras theorem in this triangle, we get
\[\begin{align}
  & {{25}^{2}}={{20}^{2}}+{{x}^{2}} \\
 & {{x}^{2}}=625-400 \\
 & {{x}^{2}}=225 \\
 & x=15 \\
\end{align}\]
Now, as one half of the other side comes out to be 15 m, so the length of this diagonal is 30 m.
Now, using the formula mentioned in the hint, we can calculate the area of this rhombus as follows
\[\begin{align}
  & =\dfrac{1}{2}\times \left( 30\times 40 \right) \\
 & =600\ {{m}^{2}} \\
\end{align}\]
Hence, the area of this rhombus is 600 square m.

Note: The students can make an error if they don’t know about the properties of a rhombus that are given in the hint as follows
The area of a rhombus is given by the formula as follows
\[=\dfrac{1}{2}\times \left( product\ of\ diagonals \right)\]
Another important fact about rhombus is that the diagonals of a rhombus bisect each other at right angles.