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

  • question_answer
    In Fig. 3, from an external point P, two tangents PT and PS are drawn to a circle with centre O and radius r. If \[OP=2r\], show that \[\angle OTS=\angle OST=30{}^\circ \].


    We have,
    Let                                \[\angle TOP=\theta \]
    In \[\Delta \,OTP,\]                      \[\cos \theta =\frac{OT}{OP}=\frac{r}{2r}=\frac{1}{2}\]
    \[\because \]                               \[\theta =60{}^\circ \]
    Hence,                          \[\angle TOS=2\theta =2\times 60{}^\circ =120{}^\circ \]
    In \[\Delta \,TOS\]
    \[\angle \,TOS+\angle OTS+\angle OST=180{}^\circ \]
                \[120{}^\circ +2\angle OTS=180{}^\circ (\because \,\angle OTS=\angle OST)\]
                            \[2\angle OTS=180{}^\circ -120{}^\circ \]
                            \[\angle OTS=30{}^\circ \]
    Hence,              \[\angle OTS=\angle OST=30{}^\circ \]              Hence Proved.


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