What are the 3 key theorems about a tangent to a circle

 


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:-   


         
    What are the 4 steps to solving geometry problems?



     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:-


    What are the 3 key theorems about a tangent to a circle



    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 Ξ”s AOP and BOP we get,
          ∵ ∠OAP=∠OBP [ ∵ Both are 90°]
         ∵  OA = OB [ ∵ Radii of the same circle]
       and,  
             OP = OP[∵Common side of the Ξ”s]
      ∴  Ξ” AOP ≌ Ξ” 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:-


    What are the 3 key theorems about a tangent to a circle?



    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:-

    What are the 3 key theorems about a tangent to a circle?




    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:-

    What are the 3 key theorems about a tangent to a circle?



    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







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