Answer
Verified
402.6k+ views
Hint: We will first find the length of each side of rhombus from the given perimeter of the rhombus. Use the property of rhombus and given diagonal to find the other diagonal of the rhombus. Hence, calculate the area of the rhombus.
Complete step-by-step answer:
The perimeter of the rhombus is 100 cm.
It is known that all sides of rhombus are equal.
Let the length of side of rhombus be $x$,
Then perimeter of rhombus is $4x$
Thus, $
4x = 100 \\
x = 25 \\
$
Hence, the length of each side of the rhombus is 25 cm.
We are given that the length of one of its diagonal is 30 cm.
In a rhombus, diagonals are perpendicular bisectors
Let diagonals bisect at $O$.
Let $AC$ be 30 cm., then the length of \[AO\] is $\dfrac{{30}}{2} = 15$cm
Also, triangle $\Delta AOB$ is a right triangle.
Apply Pythagoras theorem in $\Delta AOB$
$A{B^2} = A{O^2} + O{B^2}$
We have $AB = 25cm$ and $AO = 15cm$,
Therefore,
$
{\left( {25} \right)^2} = {\left( {15} \right)^2} + O{B^2} \\
O{B^2} = {\left( {25} \right)^2} - {\left( {15} \right)^2} \\
$
Simplifying the expression using the formula ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$
$
O{B^2} = {\left( {25} \right)^2} - {\left( {15} \right)^2} \\
O{B^2} = \left( {25 + 15} \right)\left( {25 - 15} \right) \\
O{B^2} = 40\left( {10} \right) \\
O{B^2} = 400 \\
OB = 20 \\
$
Hence, the length of the other diagonal will be twice the length $OB$, which is 40units.
Next, we will find the area of the rhombus whose formula is $\dfrac{{{d_1} \times {d_2}}}{2}$, where ${d_1}$ and ${d_2}$ are the length of diagonals.
Thus, area of rhombus is,
$\dfrac{{40 \times 30}}{2} = \dfrac{{1200}}{2} = 600c{m^2}$
Note: One must be familiar with the properties of rhombus. Also, in this question, we have used Pythagoras theorem, so one must know how to apply Pythagoras theorem in any right triangle.
Complete step-by-step answer:
The perimeter of the rhombus is 100 cm.
It is known that all sides of rhombus are equal.
Let the length of side of rhombus be $x$,
Then perimeter of rhombus is $4x$
Thus, $
4x = 100 \\
x = 25 \\
$
Hence, the length of each side of the rhombus is 25 cm.
We are given that the length of one of its diagonal is 30 cm.
In a rhombus, diagonals are perpendicular bisectors
Let diagonals bisect at $O$.
Let $AC$ be 30 cm., then the length of \[AO\] is $\dfrac{{30}}{2} = 15$cm
Also, triangle $\Delta AOB$ is a right triangle.
Apply Pythagoras theorem in $\Delta AOB$
$A{B^2} = A{O^2} + O{B^2}$
We have $AB = 25cm$ and $AO = 15cm$,
Therefore,
$
{\left( {25} \right)^2} = {\left( {15} \right)^2} + O{B^2} \\
O{B^2} = {\left( {25} \right)^2} - {\left( {15} \right)^2} \\
$
Simplifying the expression using the formula ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$
$
O{B^2} = {\left( {25} \right)^2} - {\left( {15} \right)^2} \\
O{B^2} = \left( {25 + 15} \right)\left( {25 - 15} \right) \\
O{B^2} = 40\left( {10} \right) \\
O{B^2} = 400 \\
OB = 20 \\
$
Hence, the length of the other diagonal will be twice the length $OB$, which is 40units.
Next, we will find the area of the rhombus whose formula is $\dfrac{{{d_1} \times {d_2}}}{2}$, where ${d_1}$ and ${d_2}$ are the length of diagonals.
Thus, area of rhombus is,
$\dfrac{{40 \times 30}}{2} = \dfrac{{1200}}{2} = 600c{m^2}$
Note: One must be familiar with the properties of rhombus. Also, in this question, we have used Pythagoras theorem, so one must know how to apply Pythagoras theorem in any right triangle.
Recently Updated Pages
The branch of science which deals with nature and natural class 10 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Define absolute refractive index of a medium
Find out what do the algal bloom and redtides sign class 10 biology CBSE
Prove that the function fleft x right xn is continuous class 12 maths CBSE
Find the values of other five trigonometric functions class 10 maths CBSE
Trending doubts
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Summary of the poem Where the Mind is Without Fear class 8 english CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Write an application to the principal requesting five class 10 english CBSE
What organs are located on the left side of your body class 11 biology CBSE
What is the z value for a 90 95 and 99 percent confidence class 11 maths CBSE