Question

If one of the diagonals of a rhombus is equal to its side, then the diagonals of the rhombus are in the ratio

• A. $\sqrt{3}:1$
• B. $\sqrt{2}:1$
• C. $3:1$
• D. $2:1$

Answer 1

The correct option is A $\sqrt{3}:1$

Let the side be $a$.

In a rhombus, the diagonals bisect each other at ${90}^0$.

Referring the diagram, if $PR=a$, then $OP=\frac{a}{2}$

Also, $OQ=\frac{SQ}{2}$

Since $△POQ$ is a right angled triangle,

${PQ}^2={OP}^2+{OQ}^2$

$\Rightarrow a^2={\left(\frac{a}{2}\right)}^2+{\left(\frac{SQ}{2}\right)}^2$

$\Rightarrow {\left(\frac{SQ}{2}\right)}^2=a^2-\frac{a^2}{4}=\frac{3a^2}{4}$

$\Rightarrow \left(\frac{SQ}{2}\right)=\sqrt{\frac{3a^2}{4}}=\frac{\sqrt{3}}{2}a$

$\Rightarrow SQ=\sqrt{3}a$

Hence, length of the other diagonal $=\sqrt{3}a$

Required ratio $=\frac{\sqrt{3}a}{a}=\sqrt{3}:1$