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9th Class Mathematics Circles Question Bank Circle

  • question_answer
    In the given figure below, A, B and C are three points on a circle with centre O. The tangent at C meets BA produced to T. If  \[\angle \mathbf{ATC}=\mathbf{4}{{\mathbf{0}}^{{}^\circ }}\]and \[\angle \mathbf{ACT}=\mathbf{3}{{\mathbf{8}}^{{}^\circ }}\], then what is the value of \[\angle \mathbf{AOB}\]?

    A)  \[{{78}^{{}^\circ }}\]                           

    B)  \[{{96}^{{}^\circ }}\]

    C)  \[{{102}^{{}^\circ }}\]                                  

    D)  \[{{128}^{{}^\circ }}\]

    Correct Answer: D

    Solution :

    (d): \[\angle ATC={{40}^{{}^\circ }}\text{; }\angle ACT={{38}^{{}^\circ }}\] \[\therefore \]\[\angle CAT={{180}^{{}^\circ }}-\left( {{40}^{{}^\circ }}+{{38}^{{}^\circ }} \right)\] \[=180{}^\circ -{{78}^{{}^\circ }}={{102}^{{}^\circ }}\] \[\therefore \] \[\angle OCA={{90}^{{}^\circ }}-{{38}^{{}^\circ }}\] \[={{52}^{{}^\circ }}=\angle OAC\] \[\therefore \] \[\angle OAB={{180}^{{}^\circ }}-{{102}^{{}^\circ }}-{{52}^{{}^\circ }}={{26}^{{}^\circ }}=\angle OBA\] \[\therefore \] \[\angle AOB={{180}^{{}^\circ }}-{{52}^{{}^\circ }}={{128}^{{}^\circ }}\]


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