Circles

Circles

Question

Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.


                                   OR


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

Answer

                   

Given: AB is tangent to a circle with centre O.

 

To prove: OP is perpendicular to AB.

 

Constructions: Take a point Q on AB and join OQ.

 

Proof: Since Q is a point on the tangent AB, other than the point of contact P, so Q will be outside the circle.

Let OQ intersect the circle at R.

Now OQ = OR + RQ

 OQ> OR  OQ > OP      .......[as   OR = OP] OP < OQ

Thus OP is shorter than any other segment among all and the shortest length is the perpendicular from O on AB.

 OP  AB. Hence proved.

 

                                             OR

 

                                 

Let ABCD be a quadrilateral, circumscribing a circle.

Since the tangents drawn to the circle from an external point are equal,

we have,   

AP = AS       ..........(1)

BP = BQ       ..........(2)

RC = QC       ..........(3)

DR = DS       ..........(4)

Adding, (1), (2), (3) and (4), we get

AP + PB + RC + DR = AS + BQ + QC + DS

(AP + PB ) + (RC + DR ) = (AS + DS ) + (BQ + QC)

AB + CD = AD + BC.

 

                

More Chapters from Circles