10th Class Mathematics Solved Paper - Mathematics-2016 Outside Delhi Set-I

  • question_answer Prove that the lengths of the tangents drawn from an external point to a circle are equal.


    Given, Two tangents AM and AN are drawn from point A to a circle with centre O.
    To Prove: \[AM=AN\]
    Construction: Join \[OM,ON\] and \[OA\].
    Proof: Since, \[AM\] is a tangent and \[OM\] is a radius.
    \[\therefore OM\bot AM\]
    Similarly,           \[ON\bot AN\]
    Now, in \[\Delta \,OMA\] and \[\Delta \,ONA\]
                               \[OA=OA\]                 (Common)
                              \[OM=ON\]                 (Radii of the circle)
                          \[\angle OMA=\angle ONA=90{}^\circ \]
    \[\therefore \Delta \,OMA\cong \Delta \,ONA\]         (By RHS congruence)
    Hence,              \[AM=AN\]                    Hence Proved.


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