10th Class Mathematics Triangles Question Bank Case Based (MCQs) - Triangles

  • question_answer
    What is the distance (PQ) of garden?

    A) \[10\,m\]

    B) \[11\,m\]

    C) \[12\,m\]

    D) \[13\,m\]

    Correct Answer: D

    Solution :

    Let the distance between P and Q is x m.
    Given; \[QR=4m,\] \[RS=15m,\]and \[SP=8m\]
    \[\therefore \,\,\,\,\,\,\,\,\,\,\,\,\,\,PR=PQ+QR=(4+x)m\]
    In right \[\Delta RSP,\]   \[{{(PR)}^{2}}={{(RS)}^{2}}+{{(PS)}^{2}}\]
    (By Pythagoras theorem)
    \[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,{{(x+4)}^{2}}={{(15)}^{2}}+{{(8)}^{2}}\]
    \[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,{{x}^{2}}+16+8x=225+64\,\,\,\Rightarrow \,\,{{x}^{2}}+8x-273=0\] \[\Rightarrow \,\,\,\,\,{{x}^{2}}+21x-13x-273=0\]
    (By spliting the middle term)
    \[\Rightarrow \,\,\,\,\,x(x+21)-13(x+21)=0\]
    \[\Rightarrow \,\,\,\,\,x+21=0\] or \[x-13=0\]
    But, the distance can't be negative
    \[\therefore \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x=13\]
    \[\therefore \]The required distance of garden is\[\text{13 m}\].
    So, option [d] is correct.

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