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If the diagonals of a quadrilateral are equal and bisects each other ( not at right angle ) . prove that the quadrilateral is a rectangle?
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If the diagonals of a quadrilateral are equal and bisects each other (...
Proof that the quadrilateral is a rectangle:
Given:
- The diagonals of the quadrilateral are equal
- The diagonals bisect each other (not at right angles)
Proof:
1. Diagonals are equal:
- Let the quadrilateral be ABCD with diagonals AC and BD
- Given, AC = BD
2. Diagonals bisect each other:
- Let the diagonals AC and BD bisect each other at point O
- Therefore, AO = OC and BO = OD
3. Opposite angles are equal:
- In ΔAOC and ΔBOD, AO = OC and BO = OD
- Therefore, ΔAOC ≅ ΔBOD (By SSS congruence)
- Hence, ∠A = ∠B and ∠C = ∠D
4. Consecutive angles are supplementary:
- Since opposite angles are equal, ∠A + ∠B = 180° and ∠C + ∠D = 180°
- This implies that ∠A = ∠C = 90° and ∠B = ∠D = 90°
5. Conclusion:
- From above, we can see that all angles of the quadrilateral are 90°
- Therefore, the quadrilateral ABCD is a rectangle.
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If the diagonals of a quadrilateral are equal and bisects each other ( not at right angle ) . prove that the quadrilateral is a rectangle?
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If the diagonals of a quadrilateral are equal and bisects each other ( not at right angle ) . prove that the quadrilateral is a rectangle? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about If the diagonals of a quadrilateral are equal and bisects each other ( not at right angle ) . prove that the quadrilateral is a rectangle? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the diagonals of a quadrilateral are equal and bisects each other ( not at right angle ) . prove that the quadrilateral is a rectangle?.
If the diagonals of a quadrilateral are equal and bisects each other ( not at right angle ) . prove that the quadrilateral is a rectangle? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about If the diagonals of a quadrilateral are equal and bisects each other ( not at right angle ) . prove that the quadrilateral is a rectangle? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the diagonals of a quadrilateral are equal and bisects each other ( not at right angle ) . prove that the quadrilateral is a rectangle?.
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