Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.
Answer
381.9k+ views
Hint:
In order to solve this problem, draw the diagram of quadrilateral then join its diagonals. Apply the rules of congruency of triangles among two of the triangles formed then prove the sides are equal to show it’s a rhombus. Doing this will give you the solution.
Complete step-by-step answer:
Draw a quadrilateral ABCD and join AC and BD.
Form the diagram we consider triangles AOB and AOD
It is given that diagonals bisect each other at 90 degrees.
Therefore, $\angle {\text{AOD}} = \angle {\text{AOB}}$(Right angles)
DO=OB (O is the midpoint)
AO=AO (Common side)
So, from side angle and side (SAS) rule we can say that:
$\Delta {\text{AOD}} \cong \Delta {\text{AOB}}$
Then we can say that:
AD=AB ……(1)
Now, we consider triangles AOB and BOC
It is given that diagonals bisect each other at 90 degrees.
Therefore, $\angle {\text{BOC}} = \angle {\text{AOB}}$(Right angles)
CO=OA (O is the midpoint)
BO=BO (Common side)
So, from side angle and side (SAS) rule we can say that:
$\Delta {\text{BOC}} \cong \Delta {\text{AOB}}$
Then we can say that:
BC=AB ……(2)
Now, we consider triangles COD and BOC
It is given that diagonals bisect each other at 90 degrees.
Therefore, $\angle {\text{BOC}} = \angle {\text{COD}}$(Right angles)
BO=OD (O is the midpoint)
CO=CO (Common side)
So, from side angle and side (SAS) rule we can say that:
$\Delta {\text{BOC}} \cong \Delta {\text{COD}}$
Then we can say that:
BC=CD ……(3)
So, from (1),(2) and (3) we can say that,
AB=BC=CD=AD.
So the sides are equal so it is a Rhombus.
Hence, proved.
Note – Whenever you face such types of problems always draw diagrams then proceed with the help of it. Visualizing the diagram will solve most of your queries. Here in this question it is given that the diagonals of quadrilateral bisect each other at 90 degrees, with the help of it we have used congruence of triangles to prove the sides equal then it is proved that it’s a rhombus.
In order to solve this problem, draw the diagram of quadrilateral then join its diagonals. Apply the rules of congruency of triangles among two of the triangles formed then prove the sides are equal to show it’s a rhombus. Doing this will give you the solution.
Complete step-by-step answer:
Draw a quadrilateral ABCD and join AC and BD.

Form the diagram we consider triangles AOB and AOD
It is given that diagonals bisect each other at 90 degrees.
Therefore, $\angle {\text{AOD}} = \angle {\text{AOB}}$(Right angles)
DO=OB (O is the midpoint)
AO=AO (Common side)
So, from side angle and side (SAS) rule we can say that:
$\Delta {\text{AOD}} \cong \Delta {\text{AOB}}$
Then we can say that:
AD=AB ……(1)
Now, we consider triangles AOB and BOC
It is given that diagonals bisect each other at 90 degrees.
Therefore, $\angle {\text{BOC}} = \angle {\text{AOB}}$(Right angles)
CO=OA (O is the midpoint)
BO=BO (Common side)
So, from side angle and side (SAS) rule we can say that:
$\Delta {\text{BOC}} \cong \Delta {\text{AOB}}$
Then we can say that:
BC=AB ……(2)
Now, we consider triangles COD and BOC
It is given that diagonals bisect each other at 90 degrees.
Therefore, $\angle {\text{BOC}} = \angle {\text{COD}}$(Right angles)
BO=OD (O is the midpoint)
CO=CO (Common side)
So, from side angle and side (SAS) rule we can say that:
$\Delta {\text{BOC}} \cong \Delta {\text{COD}}$
Then we can say that:
BC=CD ……(3)
So, from (1),(2) and (3) we can say that,
AB=BC=CD=AD.
So the sides are equal so it is a Rhombus.
Hence, proved.
Note – Whenever you face such types of problems always draw diagrams then proceed with the help of it. Visualizing the diagram will solve most of your queries. Here in this question it is given that the diagonals of quadrilateral bisect each other at 90 degrees, with the help of it we have used congruence of triangles to prove the sides equal then it is proved that it’s a rhombus.
Recently Updated Pages
Define absolute refractive index of a medium

Find out what do the algal bloom and redtides sign class 10 biology CBSE

Prove that the function fleft x right xn is continuous class 12 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

Find the values of other five trigonometric ratios class 10 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

Trending doubts
What is 1 divided by 0 class 8 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

What is pollution? How many types of pollution? Define it

Change the following sentences into negative and interrogative class 10 english CBSE

Why do noble gases have positive electron gain enthalpy class 11 chemistry CBSE

How fast is 60 miles per hour in kilometres per ho class 10 maths CBSE

Write an application to the principal requesting five class 10 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers
