Question

Prove that the diagonals of a rhombus are perpendicular. [3 MARKS]

Answer 1

Concept : 1 Mark
Proof : 2 Marks


Consider a rhombus $ABCD$. Let $O$ be the point of intersection of the diagonals $AC$ and $BD$

In $\Delta AOD$ and $\Delta COD,$

$OA=OC$ [Diagonal of a parallelogram bisect each other]

$OD=OD$ [Common side]

$AD=CD$ [Sides of a rhombus]

$\therefore \Delta AOD\cong \Delta COD$ [SSS congruency rule]

$\Rightarrow \angle AOD=\angle COD$......(i) [CPCT]

But, $\angle AOD+\angle COD={180}^{\circ }$ [Linear pair, since AOC is a straight line]

So, $2\angle AOD={180}^{\circ }$ [From (i)]

$\Rightarrow \angle AOD={90}^{\circ }$

$\Rightarrow$ the diagonals $AC$ and $BD$ are perpendicular.