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JEE Main & Advanced Mathematics Triangles & Properties of Triangle Question Bank Self Evaluation Test - Properties of Triangles and Height & Dstances

  • question_answer
    From an aeroplane above a straight road the angle of depression of two positions at a distance 20 m apart on the road are observed to be \[30{}^\circ \] and \[45{}^\circ \]. The height of the aeroplane above the ground is:

    A) \[10\sqrt{3}\,m\]

    B) \[10(\sqrt{3}-1)m\]

    C) \[10(\sqrt{3}+1)m\]

    D) 20 m

    Correct Answer: C

    Solution :

    [c]
    In \[\Delta ABC,\tan 45{}^\circ =\frac{AB}{BC}=\frac{h}{x};1=\frac{h}{x}\]
    \[h=x\]                          (i)
    In \[\Delta ABD,\]
    \[\tan 30{}^\circ =\frac{AB}{BD};\]       \[\frac{1}{\sqrt{3}}=\frac{h}{x+20}\]
    \[x+20=\sqrt{3}h;\]       \[h+20=\sqrt{3}h\]
    \[20=\left( \sqrt{3}-1 \right)h;\]     \[h=\frac{20}{\sqrt{3}-1}\]
    \[=\frac{20}{\sqrt{3}-1}\times \frac{\sqrt{3}+1}{\sqrt{3}+1}\]
    \[=\frac{20(\sqrt{3}+1)}{2}=10(\sqrt{3}+1)m\]
    Hence the height is \[10(\sqrt{3}+1)m\]


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