Ex.7.2 Q10 Coordinate Geometry Solution - NCERT Maths Class 10

Go back to  'Ex.7.2'


Find the area of a rhombus if its vertices are \((3, 0)\), \((4, 5)\), \((-1, 4)\) and \((-2, -1)\) taken in order.

[Hint: Area of a rhombus \(=\) (product of its diagonals)]


Text Solution


A rhombus has all sides of equal length and opposite sides are parallel.

What is the known?

The \(x\) and \(y\) co-ordinates of the vertices of the rhombus.

What is the unknown?

The area of the rhombus


From the Figure,


  • Let \(A(3, 0)\), \(B(4, 5)\), \(C(-1, 4)\) and \(D(-2, -1)\) are the vertices of a rhombus \(ABCD\).

We know that the distance between the two points is given by the Distance Formula,

\[\begin{align}\sqrt {{{\left( {{{{x}}_1} - {{{x}}_2}} \right)}^2} + {{\left( {{{{y}}_1} - {{\text{y}}_2}} \right)}^2}} & & ...\;{\text{Equation}}\;{\text{(1)}}\end{align}\]

Therefore, distance between \(A\;(3, 0)\) and \(C \;(-1, 4)\) is given by

Length of diagonal

\[\begin{align} AC &= \sqrt {{{[3 - ( - 1)]}^2} + {{(0 - 4)}^2}}  \\ &= \sqrt {16 + 16} = 4\sqrt 2 \end{align}\]

Therefore, distance between B (4, 5) and D (-2, -1) is given by

Length of diagonal

\[\begin{align} BD &= \sqrt {{{[4 - ( - 2)]}^2} + \left( {5 - {{( - 1)}^2}} \right.}   \\& = \sqrt {36 + 36} \\&= 6\sqrt 2 \end{align}\]

 \[\begin{align} \text{Area of the rhombus } ABCD &= \frac{1}{2} \times \rm{(Product\; of \;lengths\; of \;diagonals)} \\&= \frac{1}{2} \times {\text{AC}} \times {\text{BD}}\end{align}\]

Therefore, area of rhombus

\[\begin{align}ABCD &= \frac{1}{2} \times 4\sqrt 2 \times 6\sqrt 2 \\ &= 24 \;\rm square\; units\end{align}\]


Frequently Asked Questions

What are Class 10 NCERT Exemplars?
While getting good scores in school tests is a desirable outcome, it is not a reliable indicator of how strong your child’s math foundation really is. Many students who score well in school exams in their earlier years, might struggle with math in higher grades because of a weak foundation. At Cuemath, we evaluate your child’s grasp of math fundamentals, and take corrective actions immediately. Also, your child may have limited exposure in their school, and in most cases, may not feel challenged to learn more. Cuemath's customised learning plan ensures your child is challenged with varied difficulty levels of questions at every stage.
What is the difference between CBSE and NCERT syllabus for Class 10?
How will Class 10 NCERT books help in exam preparation?
How will Class 10 NCERT books help you understand basic math concepts?
Which is the best video solution for the class 10 maths NCERT?