Question

If the sum of the diagonal of a rhombus is 10 cm and its perimeter is $4\sqrt{1}3$ cm, then the lengths of its diagonals are :

• A. $5,5$
• B. $6,4$
• C. $7,3$
• D. $8,2$

Answer 1

Answer: (B)

$6,4$

Let the side of rhombus is a and length of diagonals cut each other length x and y

We know that all sides of rhombus are equal and the diagonal intersect perpendicular each other

$\therefore 4a=4\sqrt{13}$ (Given perimeter )

$\Rightarrow a=\sqrt{13}$

$2x+2y=10$(given diagonal length)

$\Rightarrow x+y=5$

$\Rightarrow x=5-y$..............(1)

In triangle OCD

$x^2+y^2=13$......................(2)

Put the value of x as per (1)

$(5-y{)}^2+y^2=13$

$\Rightarrow 25-10y+y^2+y^2=13$

$\Rightarrow 2y^2-10y+25-13=0$

$\Rightarrow 2y^2-10y+12=0$

$\Rightarrow y^2-3y-2y+6=0$

$\Rightarrow y(y-3)-2(y-3)=0$

$\Rightarrow (y-3)(y-2)=0$

Then $y=2$ or$y=3$

Put value $y=2$

Then $x=5-2=3$

Then length of diagonals are

$6$, $4$