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Maths / Circles / Length of Tangents to a Circle
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Length of Tangents to a Circle

Length of Tangents to a Circle: it is the distance of the point P to the point of contact on the circle that is point A and Point B. Or in other words PA and PB are the lengths on tangent from point P to the circle

Theorem: The length of tangents drawn from an external point to a circle are equal.

Given :A circle with centre O. PA and PB to tangents from point P.

To Prove: PA = PB 

Proof:  Join OA an OB 

 In  large Delta OAP  and  large dpi{120} large Delta OBP

                OA = OB                          [radii of the same circle]

                OP = OP                          [ common ]

   large angle OAP =angle OBP   [ each large 90^{circ}]

Thus large Delta OAP cong Delta OBP    [ RHS criterion]

         PA = PB  [CPCT]

Thus, lengths of tangents from an external point to a circle are equal.

Corollary: Theline joining the centre of the circle and the external point i.e. OP is the angle bisector of angle AOB and angleAPB.

         As    large Delta OAPcong Delta OBP,

         large angle APO=angle BPO    [By CPCT]

large Rightarrow  large PO  is the angle bisectors of large angle APB 

Similarly OP is the angle bisector of angle AOB

ILLUSTRATION : A quadrilateral ABCD is drawn to circumscribe a circle. Prove that      AB + CD =AD + BC

Solution    A quadrilateral ABCD circumscribes a cricle. Since the lengths of tangents from an external point to a circle are equal, we get

     AP = AS                    [ lengths of tangents from point A ]

     BP = BQ                   [ lengths of tangents from point B ]

     CR = CQ                  [ lengths of tangents from point C ]

     DR = DS                  [ lengths of tangents from point D ]

Adding the above expressions, we get

      AP + BP + CR + DR = AS + BQ + CQ + DS

large Rightarrow  ( AP + BP ) + ( CR + DR ) = ( AS + DS ) + ( BQ + CQ )

large Rightarrow              AB + CD = AD + BC

Hence Proved

 

 


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