Show that if the diagonals of a quadrilateral bisect each other at right angles,then it is a rhombus
Answers
Hi mate here is the answer:--✍️✍️
Question:-✔️✔️
Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.
Solution:-✔️✔️
To Prove:
If diagonals of a quadrilateral bisect at 90º, it is a rhombus.
Definition of Rhombus:
A parallelogram whose all sides are equal.
Given:
Let ABCD be a quadrilateral whose diagonals bisect at 90º.
In ΔAOD and ΔCOD,
In ΔAOD and ΔCOD,OA = OC (Diagonals bisect each other)
In ΔAOD and ΔCOD,OA = OC (Diagonals bisect each other)∠AOD = ∠COD (Given)
In ΔAOD and ΔCOD,OA = OC (Diagonals bisect each other)∠AOD = ∠COD (Given)OD = OD (Common)
∆AOD congruent ∆ ΔCOD (By SAS congruence rule)
AD = CD ..................(1)
Similarly,
AD = AB and CD = BC ..................(2)
From equations (1) and (2),
AB = BC = CD = AD
Since opposite sides of quadrilateral
ABCD are equal, it can be said that ABCD is a parallelogram. Since all sides of a parallelogram ABCD are equal, it can be said that
ABCD is a rhombus
Hence, Proved.
Hope it helps you ❣️☑️☑️
Answer:
Hi mate here is the answer:--✍️✍️
Question:-✔️✔️
Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.
Solution:-✔️✔️
To Prove:
If diagonals of a quadrilateral bisect at 90º, it is a rhombus.
Definition of Rhombus:
A parallelogram whose all sides are equal.
Given:
Let ABCD be a quadrilateral whose diagonals bisect at 90º.
In ΔAOD and ΔCOD,
In ΔAOD and ΔCOD,OA = OC (Diagonals bisect each other)
In ΔAOD and ΔCOD,OA = OC (Diagonals bisect each other)∠AOD = ∠COD (Given)
In ΔAOD and ΔCOD,OA = OC (Diagonals bisect each other)∠AOD = ∠COD (Given)OD = OD (Common)
∆AOD congruent ∆ ΔCOD (By SAS congruence rule)
AD = CD ..................(1)
Similarly,
AD = AB and CD = BC ..................(2)
From equations (1) and (2),
AB = BC = CD = AD
Since opposite sides of quadrilateral
ABCD are equal, it can be said that ABCD is a parallelogram. Since all sides of a parallelogram ABCD are equal, it can be said that
ABCD is a rhombus
Hence, Proved.
Hope it helps you ❣️☑️☑️
Step-by-step explanation: