10th Class Mathematics Solved Paper - Mathematics-2016 Delhi Term-II Set-I

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


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


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