10th Class Mathematics Solved Paper - Mathematics-2018

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


    Given: a circle with centre O on which two tangents PM and PN are drawn from an external point P.
    To prove:
                   PM = PN
    Construction: Join OM, ON and OP.
    Proof: Since tangent and radius are perpendicular at point of contact,
    \[\therefore \]    \[\angle OMP=\angle ONP=90{}^\circ \]
    In \[\Delta \text{ }POM\]and \[\Delta \text{ }PON\],
                     \[OM=ON\]                                      (Radii)
                \[\angle OMP=\angle ONP\]
                      \[PO=OP\]                    (Common)
    \[\therefore \]      \[\Delta \,OMP\cong \Delta \,ONP\]                            (RHS cong.)
    \[\therefore \]        \[PM=PN\]                              (CPCT)
                                                                      Hence Proved.     


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