Question

Given a parallelogram whose acute angle is $60^{\circ }$, if the squares of length of the diagonals are in ratio $1:3$ then $\frac{a}{b}$ is (where $a,b$ are the length of the sides)

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (A)

Applying cosine rule in $△ABD$ and $△ABC$,
we get $2a^2+2b^2=d_1^2+d_2^2$,
where $d_1,d_2$ are lengths of diagonals
$\left(d_1<d_2\right)$ given $d_2^2=3d_1^2$
$\Rightarrow a^2+b^2=2d_1^2$
Also $ab=d_1^2$ ...(1)
$\Rightarrow a^2+b^2+2ab=4d_1^2$
$\Rightarrow (a+b{)}^2=4d_1^2$
$\Rightarrow a+b=2d_1$ ...(2)
Using (1) and (2), we get $a=b$ i.e., $\frac{a}{b}=1$.