Name: Theorems on a Parallelogram
Uploaded: 2019-07-19
Duration: 12 min 6 s
Description: Learn the theorems on quadrilaterals, such as properties related to the diagonal of a parallelogram and the opposite sides and angles of a parallelogram etc.

Question

Show that the diagonals of a rhombus divide it into four congruent triangles.

Accepted Answer

Let us take, $ABCD$ is a rhombus.

In $ΔAOD$ and $ΔCOB$ ,

$\Rightarrow AO = OC$ [diagonal of rhombus bisect each other]

$\Rightarrow OD = OB$ [diagonal of rhombus bisect each other]

$\Rightarrow AD = BC$ [all sides of rhombus are equal]

$∴ Δ A O D ≅ Δ C O B$ (By $S . S . S$ property).

In $∆ AOB$ and $ΔCOD$ ,

$\Rightarrow AO = OC$ [diagonal of rhombus bisect each other]

$\Rightarrow OD = OB$ [diagonal of rhombus bisect each other]

$\Rightarrow CD = BA$ [all sides of rhombus are equal]

$∴ ΔAOB ≅ ΔCOD$ .

In $ΔAOB$ and $ΔAOD$

$\Rightarrow AO = AO$ [Common in both triangles]

$\Rightarrow OD = OB$ [diagonal of rhombus bisect each other]

$\Rightarrow AD = AB$ [all sides of rhombus are equal]

$∴ ΔAOB ≅ ΔAOD$ .

Similarly, $∆ COD ≅ ∆ COB$ .

$∴$ All four triangles divide by diagonals of rhombus are congruent.

Show that the diagonals of a rhombus divide it into four congruent triangles.

Important Questions on Quadrilaterals

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