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PA and PB are two tangents drawn from an external point P to a circle with centre O where the points A and B are the points of contact. The quadrilateral OAPB must be (SSC Sub. Ins. 2012)
- a)a square
- b)concyclic
- c)a rectangle
- d)a rhombus
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
PA and PB are two tangents drawn from an external point P to a circle ...
OAPB is concyclic because ∠A + ∠B = 180°
& ∠O + ∠P = 180°
Most Upvoted Answer
PA and PB are two tangents drawn from an external point P to a circle ...
Given:
PA and PB are two tangents drawn from an external point P to a circle with center O, where the points A and B are the points of contact.
To prove:
The quadrilateral OAPB is concyclic.
Proof:
Definition:
Concyclic points are points that lie on the same circle.
Construction:
Let's draw the diagram to understand the given information.
Proof by contradiction:
Assume that the quadrilateral OAPB is not concyclic.
1. Angle OPA:
Since PA is a tangent to the circle, angle OPA is a right angle. (Tangent is perpendicular to the radius at the point of contact)
2. Angle OPB:
Similarly, since PB is a tangent to the circle, angle OPB is also a right angle.
3. Angle OAP:
Since triangle OPA is a right-angled triangle, angle OAP is the complement of angle OPA. Therefore, angle OAP is also a right angle.
4. Angle OB:
Since triangle OPB is a right-angled triangle, angle OPB is the complement of angle OPB. Therefore, angle OPB is also a right angle.
5. Angle APB:
Since angle OPA and angle OPB are right angles, the sum of the angles in triangle APB is 180 degrees. Therefore, angle APB is a straight angle.
6. Conclusion:
From the above steps, we can see that all the angles in the quadrilateral OAPB are right angles (OPA, OPB, OAP, and OPB).
If all the angles in a quadrilateral are right angles, then the quadrilateral is a rectangle.
Therefore, the quadrilateral OAPB is a rectangle.
Hence, option 'c' is the correct answer.
PA and PB are two tangents drawn from an external point P to a circle with center O, where the points A and B are the points of contact.
To prove:
The quadrilateral OAPB is concyclic.
Proof:
Definition:
Concyclic points are points that lie on the same circle.
Construction:
Let's draw the diagram to understand the given information.
Proof by contradiction:
Assume that the quadrilateral OAPB is not concyclic.
1. Angle OPA:
Since PA is a tangent to the circle, angle OPA is a right angle. (Tangent is perpendicular to the radius at the point of contact)
2. Angle OPB:
Similarly, since PB is a tangent to the circle, angle OPB is also a right angle.
3. Angle OAP:
Since triangle OPA is a right-angled triangle, angle OAP is the complement of angle OPA. Therefore, angle OAP is also a right angle.
4. Angle OB:
Since triangle OPB is a right-angled triangle, angle OPB is the complement of angle OPB. Therefore, angle OPB is also a right angle.
5. Angle APB:
Since angle OPA and angle OPB are right angles, the sum of the angles in triangle APB is 180 degrees. Therefore, angle APB is a straight angle.
6. Conclusion:
From the above steps, we can see that all the angles in the quadrilateral OAPB are right angles (OPA, OPB, OAP, and OPB).
If all the angles in a quadrilateral are right angles, then the quadrilateral is a rectangle.
Therefore, the quadrilateral OAPB is a rectangle.
Hence, option 'c' is the correct answer.
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PA and PB are two tangents drawn from an external point P to a circle with centre O where the points A and B are the points of contact. The quadrilateral OAPB must be (SSC Sub. Ins. 2012)a)a squareb)concyclicc)a rectangled)a rhombusCorrect answer is option 'B'. Can you explain this answer?
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