Name: Illustration Based on the SAS Criterion
Uploaded: 2019-07-19
Duration: 4 min 36 s
Description: One of the easiest ways to learn a concept is through a visual illustration. To learn how the SAS congruence criterion is applied in mathematical problems, w...

Question

Prove that the diagonals of a rhombus bisect each other at right angles.

Accepted Answer

Let $A B C D$ is a rhombus.

We know that, all the sides of rhombus are equal and opposite side are parallel.

So, $A B ∥ C D, A D ∥ B C$ and $A B = B C = C D = A D$ .

$A C and B D$ are the diagonals of the rhombus.

We know that, alternate angles made by transversal of parallel lines are equal.

So, $∠ B A O = ∠ D C O, ∠ A B O = C D O, ∠ D A O = ∠ B C O, ∠ C B O = ∠ A D O$ .

Now, in $∆ A O B and ∆ C O D$ ,

$∠ B A O = ∠ D C O$

$∠ A B O = ∠ C D O$

$A B = C D$

So, $∆ A O B ≅ ∆ C O D$ (by A.S.A. congruency)

Now, corresponding sides of congruent triangles are equal.

So, $A O = C O$

And $O B = O D$

So, diagonals of rhombus bisects each other.

Now, in $∆ A O B and ∆ A O D$ ,

$O B = O D$

$A O = A O$ (common side)

$A B = A D$

So, $∆ A O B ≅ ∆ A O D$ (By S.S.S. congruency)

So, $∠ A O B = ∠ A O D$ (corresponding angles of congruent triangles)

Now, we know that, sum of linear pair angles is $180 °$ .

So, $∠ A O B + ∠ A O D = 180 °$

$\Rightarrow ∠ A O B + ∠ A O B = 180 °$

$\Rightarrow 2 ∠ A O B = 180 °$

$\Rightarrow ∠ A O B = 90 °$

So, $∠ A O D = 90 °$

So, the diagonals of a rhombus bisect each other at right angles.

Hence, proved.

Prove that the diagonals of a rhombus bisect each other at right angles.

Important Questions on Triangles

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