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In the figure PA perpendicular AB ,QB perpendicular AB and PA=QB. If PQ intersects AB at O , Show that O is the mid point of AB as well as that of PQ.?
Verified Answer
In the figure PA perpendicular AB ,QB perpendicular AB and PA=QB. If P...
In ∆AOP and ∆ BOQ
AP= BQ. (given)
angle PAO= Angle QBO. (each 90degree)
Angle AOP=angle BOQ. (V.O.A)
therefore ∆AOP =~ ∆BOQ.
Therefore OA=OB
Therefore O is midpoint of AB.
This question is part of UPSC exam. View all Class 9 courses
Most Upvoted Answer
In the figure PA perpendicular AB ,QB perpendicular AB and PA=QB. If P...
To prove that O is the midpoint of AB and PQ, we will use the given information that PA is perpendicular to AB, QB is perpendicular to AB, and PA is equal to QB.
Proof:
Step 1: Show that O is the midpoint of AB
Since PA is perpendicular to AB and QB is perpendicular to AB, we can conclude that PAO and QBO form right angles.
In triangle PAO and triangle QBO,
- PA is equal to QB (given)
- AO is equal to BO (common side)
- PAO and QBO are right angles
By the hypotenuse-leg congruence theorem, we can conclude that triangle PAO is congruent to triangle QBO.
Therefore, the corresponding sides of these triangles are congruent, which means that AO is equal to BO.
Hence, O is the midpoint of AB.
Step 2: Show that O is the midpoint of PQ
To prove that O is the midpoint of PQ, we need to show that PO is equal to OQ.
Since PA is equal to QB, we can conclude that triangle PAQ and triangle QBQ are congruent by the side-side-side congruence theorem.
Therefore, the corresponding sides of these triangles are congruent, which means that PO is equal to OQ.
Hence, O is the midpoint of PQ.
Conclusion:
From the above proof, we can conclude that O is the midpoint of both AB and PQ.
Proof:
Step 1: Show that O is the midpoint of AB
Since PA is perpendicular to AB and QB is perpendicular to AB, we can conclude that PAO and QBO form right angles.
In triangle PAO and triangle QBO,
- PA is equal to QB (given)
- AO is equal to BO (common side)
- PAO and QBO are right angles
By the hypotenuse-leg congruence theorem, we can conclude that triangle PAO is congruent to triangle QBO.
Therefore, the corresponding sides of these triangles are congruent, which means that AO is equal to BO.
Hence, O is the midpoint of AB.
Step 2: Show that O is the midpoint of PQ
To prove that O is the midpoint of PQ, we need to show that PO is equal to OQ.
Since PA is equal to QB, we can conclude that triangle PAQ and triangle QBQ are congruent by the side-side-side congruence theorem.
Therefore, the corresponding sides of these triangles are congruent, which means that PO is equal to OQ.
Hence, O is the midpoint of PQ.
Conclusion:
From the above proof, we can conclude that O is the midpoint of both AB and PQ.
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In the figure PA perpendicular AB ,QB perpendicular AB and PA=QB. If PQ intersects AB at O , Show that O is the mid point of AB as well as that of PQ.?
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In the figure PA perpendicular AB ,QB perpendicular AB and PA=QB. If PQ intersects AB at O , Show that O is the mid point of AB as well as that of PQ.? 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 the figure PA perpendicular AB ,QB perpendicular AB and PA=QB. If PQ intersects AB at O , Show that O is the mid point of AB as well as that of PQ.? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In the figure PA perpendicular AB ,QB perpendicular AB and PA=QB. If PQ intersects AB at O , Show that O is the mid point of AB as well as that of PQ.?.
In the figure PA perpendicular AB ,QB perpendicular AB and PA=QB. If PQ intersects AB at O , Show that O is the mid point of AB as well as that of PQ.? 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 the figure PA perpendicular AB ,QB perpendicular AB and PA=QB. If PQ intersects AB at O , Show that O is the mid point of AB as well as that of PQ.? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In the figure PA perpendicular AB ,QB perpendicular AB and PA=QB. If PQ intersects AB at O , Show that O is the mid point of AB as well as that of PQ.?.
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