Show that the diagonals of a rhombus are perp | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(N/A) Consider the rhombus $ABCD$ (see Fig.).You know that $AB = BC = CD = DA$ (by definition of a rhombus).Now,in $\Delta AOD$ and $\Delta COD$,$OA =

Vedclass · Accepted Answer

Vedclass · Answer

(N/A) Consider the rhombus $ABCD$ (see Fig.).
You know that $AB = BC = CD = DA$ (by definition of a rhombus).
Now,in $\Delta AOD$ and $\Delta COD$,
$OA = OC$ (Diagonals of a parallelogram bisect each other).
$OD = OD$ (Common side).
$AD = CD$ (Sides of a rhombus are equal).
Therefore,$\Delta AOD \cong \Delta COD$ ($SSS$ congruence rule).
This gives,$\angle AOD = \angle COD$ $(CPCT)$.
But,$\angle AOD + \angle COD = 180^{\circ}$ (Linear pair).
So,$2 \angle AOD = 180^{\circ}$.
Or,$\angle AOD = 90^{\circ}$.
Thus,the diagonals of a rhombus are perpendicular to each other.

Show that the diagonals of a rhombus are perpendicular to each other.

Similar Questions

$ABCD$ is a trapezium in which $AB \parallel CD$ and $AD = BC$ (see Fig). Show that $\angle A = \angle B$.

In parallelogram $ABCD$,$AB = 12 \, cm$. Altitudes $DM$ and $DN$ correspond to bases $AB$ and $BC$ respectively. If $DM = 5 \, cm$ and $DN = 6 \, cm$,then find the length of $BC$ in $cm$.

$ABCD$ is a rectangle and $P$,$Q$,$R$ and $S$ are mid-points of the sides $AB$,$BC$,$CD$ and $DA$ respectively. Show that the quadrilateral $PQRS$ is a rhombus.

$ABCD$ is a parallelogram and $AP$ and $CQ$ are perpendiculars from vertices $A$ and $C$ on diagonal $BD$ (see figure). Show that $AP = CQ$.

Prove that a diagonal of a parallelogram divides it into two congruent triangles.

Vedclass Test Series

Exam Paper Generator

Online Exam Module