What are the 3 key theorems about a tangent to a circle?
Today in this article I will explain a very important topics,which is circle and its various theorems.
Contents:-
*Prove that radius of a circle is perpendicular on the tangent at its point of contact.
Answer:-
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Let PQ is a tangent on a circle with cener 'O' at a point 'R' and OR is the radius of the circle.
To prove:- OR⊥PQ
Construction:- 'M' is an another point taken on 'PQ' outside the circle
Proof:-
If M would inside the circle then PQ would be a secant not a tangent, but in this theorem PQ must be a tangent, So clearly we can see that the distance of 'M' from the center of the circle 'O' is greater than the distance of point 'R' from the center 'O'
i.e; OM> OR, and this is universal true for any pont lying outside the circle and which is on PQ. So 'OR' is the least smaller distance from center 'O'
∵ We know that line drawn from a fixed point on another line the shortest vertical line is the the perpendicular.
So, OR⊥ PQ Proved.
*Prove that the length of the tangents drawn from an external points on a circle are equal and they also produces equal angles at the center.
Answer:-
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Let P be an external point, PA and PB are two drawn tangents on the circle 'O' at the points 'A' and 'B' respectively.
To prove:-
(i) PA = PB
(ii) <POA = <POB
Construction:- 'OP' is joined.
Proof:- ∵ OA is the radius of the circle and PA is a tangent at point A of the circle,
∴ OA⊥ PA
∴ ∠OAP = 90°
Similarly,
<OBP = 90°
Now,
In between Δs AOP and BOP we get,
∵ ∠OAP=∠OBP [ ∵ Both are 90°]
∵ OA = OB [ ∵ Radii of the same circle]
and,
OP = OP[∵Common side of the Δs]
∴ Δ AOP ≌ Δ BOP
∴ AP = BP
* What is a Cyclic quadrilateral?
Answer:-
A quadrilateral whose all four angular points touches the arc of a circle is called a Cyclic quadrilateral.
* Prove that sum of opposite angles of a cyclic quadrilateral is 180°
Answer:-
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Let 'ABCD' is a cyclic quadrilateral.
To prove:- <ABC + <ADC = 180°
and <BAD + < BCD = 180°
Construction:- OA and OC are joined.
Proof:- ∵∠ABC and obtuse ∠AOC are the angles at the circumference and at the center standing on the same arc ADC
∴∠ABC = ½ obtuse <AOC…..(i) [∵angle at the circumference is half the angle at the center standing on the same arc]
Again,
<ADC and reflex <AOC are the angles at the circumference and at the center standing on the same arc ABC
∴ ∠ADC =½ reflex <AOC…..(2) [angle at the circumference is half the angle at the center standing on the same arc]
(1)+(2)
<ABC+<ADC=½(obtuse<AOC+reflex<AOC)
=><ABC+<ADC=½ ×360°
=><ABC+<ADC =180° Proved.
*Prove that semi circle angle is 90°
Answer:-
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Let, AB is a diameter of a circle with center'O' ∴∠ACB is an angle of semi circle
To prove:- <ACB = 90°
Proof:-
∵∠ACB and ∠AOB are the angles at the circumference and at the center respectively standing on the same arc AB
∴ ∠ACB = ½ <AOB [∵ angle at the circumference is half the angle at the center standing on the same arc]
=> <ACB = ½ ×180° [ ∵∠AOB is a straight angle.∴∠AOB=180°]
=><ACB = 90° Proved.
*Prove that angles at the circumference of a circle are equal.
Answer:-
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Let,
<ACB and <ADB are the angles at the circumference and <AOB at the center standing on the same arc APB
To prove:- <ACB = <ADB
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FREQUENTLY ASKED QUESTIONS:-
How many tangents can be drawn in a circle?
Answer:- Infinite number of tangents can be drawn in a circle.
What is the relation between a tangent and a radius of a circle ?
Answer :- Relation is = r ⊥ t where, r = radius of the circle and t = tangent of the circle.
What is tangent of a circle?
Answer :- Any line which touches the circle at a point, then that line is called the tangent of that circle.
In how many points a tangent touches a circle?
Answer:- One.
What is secent of a circle?
Answer:- Any line which touches the circle
at two different points, then that line is called the secent of that circle.
In how many points a secent touches a circle?
.
Answer:- Two
How many tangents can be drawn in a circle?
Answer:- Infinite number of tangents can be drawn in a circle.
What is the relation between a tangent and a radius of a circle ?
Answer :- Relation is = r ⊥ t where, r = radius of the circle and t = tangent of the circle.
What is tangent of a circle?
Answer :- Any line which touches the circle at a point, then that line is called the tangent of that circle.
In how many points a tangent touches a circle?
Answer:- One.
What is secent of a circle?
Answer:- Any line which touches the circle at two different points, then that line is called the secent of that circle.
In how many points a secent touches a circle?
. Answer:- Two
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