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This content is part of NCERT Solutions for Class 10 Maths Chapter 10 Circles

9.4        THEORM :

 Statement :       Lengths of two tangents drawn from an external point to a circle are equal.

Given:              AP and AQ are two tangents drawn from a point A to a circle C (O, r).

 To prove :        AP = AQ

 Construction :  Join OP, OQ and OA.

 Proof :              In AOQ and APO    

                       [Tangent at any point of a circle is perp. to radius through the point of contact]

AO = AO           [Common]

OQ = OP          [Radius]

So, by R.H.S. criterion of congruency

This content is part of NCERT Solutions for Class 10 Maths Chapter 6 Triangles

THALES THEOREM (BASIC PROPROTIONALITY THEOREM) :

 Statement :          If a line is drawn parallel to one side of a triangle to intersect the other sides in distinct points, then the other two sides are divided in the same ratio.

 Given:                   A triangle ABC in which a line parallel to side

BC intersects other two sides AB and AC at D and

E respectively.

To Prove :            

Construction :    Join BE and CD and draw DMAC and ENAB.

Proof :                   Area of DADE (=base × height) = AD × EN.

                               Area of DADE is denoted as are (ADE)

                               So,          ar(ADE) = DB × EN

                               And       ar(BDE) = DB × EN ,

                               Therefore,                  ….(i)

                               Similarly,             ar(ADE =AE × DM and ar(DEC =EC × DM.

                               And                                            ......(ii)

Note that D BDE and D DEC are on the same base DE and between the two parallel lines BC and DE.

                               So,                          ar(BDE) = ar(DEC)                            .....(iii)