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In quadrilateral ABCD, BN and DM are drawn perpendicular to AC. Such that BN = DM. Prove
that O is mid-point of BD.?
that O is mid-point of BD.?
Most Upvoted Answer
In quadrilateral ABCD, BN and DM are drawn perpendicular to AC. Such t...
Given:
- Quadrilateral ABCD
- BN and DM are perpendicular to AC
- BN = DM
To Prove:
- O is the midpoint of BD
Proof:
Step 1: Construction
- Draw the perpendicular bisectors of BN and DM and let them intersect at point O.
Step 2: Proof of BN = DM
- Since BN and DM are perpendicular to AC, they are parallel to each other.
- Thus, BN and DM are opposite sides of a parallelogram.
- In a parallelogram, opposite sides are equal.
- Therefore, BN = DM.
Step 3: Proof of BO = DO
- In triangle BON and DOM,
- BN = DM (given)
- ON = OM (perpendicular bisector)
- Angle BNO = Angle DMO (90 degrees)
- By the Side-Angle-Side (SAS) congruence criterion, we can conclude that triangle BON is congruent to triangle DOM.
- Therefore, BO = DO.
Step 4: Proof of AO = CO
- In triangle AON and CON,
- ON = ON (common side)
- AO = CO (perpendicular bisector)
- Angle AON = Angle CON (90 degrees)
- By the Side-Angle-Side (SAS) congruence criterion, we can conclude that triangle AON is congruent to triangle CON.
- Therefore, AO = CO.
Step 5: Proof of O is the midpoint of BD
- In quadrilateral ABCD, we have shown that BO = DO and AO = CO.
- Therefore, O is the midpoint of both BD and AC.
Conclusion:
- Therefore, we have proved that O is the midpoint of BD in quadrilateral ABCD.
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In quadrilateral ABCD, BN and DM are drawn perpendicular to AC. Such that BN = DM. Provethat O is mid-point of BD.?
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In quadrilateral ABCD, BN and DM are drawn perpendicular to AC. Such that BN = DM. Provethat O is mid-point of BD.? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about In quadrilateral ABCD, BN and DM are drawn perpendicular to AC. Such that BN = DM. Provethat O is mid-point of BD.? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In quadrilateral ABCD, BN and DM are drawn perpendicular to AC. Such that BN = DM. Provethat O is mid-point of BD.?.
In quadrilateral ABCD, BN and DM are drawn perpendicular to AC. Such that BN = DM. Provethat O is mid-point of BD.? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about In quadrilateral ABCD, BN and DM are drawn perpendicular to AC. Such that BN = DM. Provethat O is mid-point of BD.? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In quadrilateral ABCD, BN and DM are drawn perpendicular to AC. Such that BN = DM. Provethat O is mid-point of BD.?.
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