slove it fast plz 3rd and 2nd ☺️
Class 9th
Answers
Question 2 :
›»› If the diagonals of a parallelogram are equal, then show that it is a rectangle.
Given :
- ABCD is a parallelogram.
- AC = DB
To Prove :
- ABCS is a rectangle.
Proof :
In ∆ABC and ∆DCB,
⪼ AC = DB [Given]
⪼ BC = BC [Common]
⪼ AB = DC [Opposite sides of a parallelogram are equal]
Therefore, ∆ABC = ∆DCB [By SSS-congruency]
So, ∠ABC = ∠DCB [CPCT]
As we know the sum of the measures of angles on the same side of transversal is 180°
→ ∠ABC + ∠DCB = 180°
→ ∠ABC + ∠ABC = 180°
→ 2∠ABC = 180°
→ ∠ABC = 180/2
→ ∠ABC = 90°
Since ABCD is a parallelogram and one of its interior angles is 90° So, ABCD is a rectangle.
Hence, Proved !
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Question 3 :
›»› Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.
Given :
- OA = OC
- OB = OD
- ∠AOB = ∠BOC = ∠OCD = ∠ODA = 90°
To Prove :
- ABCD is a rhombus.
- AB = BC = CD = AD
Proof :
In ΔAOB and ΔCOB,
⪼ OA = OC [Given]
⪼ OB = OB [Common]
⪼ ∠AOB = ∠COB [Opposite sides of a parallelogram are equal]
Therefore, ΔAOB ≅ ΔCOB [by SAS-Congruency]
So, AB = BC [CPCT]
Similarly ,
We can prove,
→ BC = CD
→ CD = AD
→ AD = AB
→ AB = BC = CD = AD
Opposites sides of a quadrilateral are equal hence ABCD is a parallelogram.
A rhombus is also a parallelogram so it is also a rhombus.
Hence Proved !
