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In a triangle ABC, a=7, b=8, c=9, BD is the median and BE is the altitude from the vertex B, then BD is equals to _____

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answer is D.

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Detailed Solution

The diagram of the above equation is,

https://www.vedantu.com/question-sets/cc62530c-ac02-459d-961d-66a2daa19a231584078479262730930.png

In ΔABC, the cosine formula for A is given by,

cosA = \frac{b^{2} + c^{2} - a^{2}}{2 bc}

\Rightarrow cosA = \frac{8^{2} + 9^{2} - 7^{2}}{2 \times 8 \times 9}

\Rightarrow cosA = \frac{64 + 81 - 49}{144}

\Rightarrow cosA = \frac{96}{144}

\Rightarrow cosA = \frac{2}{3}

.............….. (1)
In ΔABD, the cosine formula for A is given by,

cosA = \frac{A B^{2} + A D^{2} - B D^{2}}{2 \times AB \times AD}

\Rightarrow cosA = \frac{9^{2} + 4^{2} - B D^{2}}{2 \times 9 \times 4} \Rightarrow cosA = \frac{81 + 16 - B D^{2}}{72} \Rightarrow cosA = \frac{97 - B D^{2}}{72} . . . . . . . . . . . . . . . . \dots . . (2)

Compare the equation (1) and (2),

\Rightarrow \frac{97 - B D^{2}}{72} = \frac{2}{3}

Cross-multiply the terms,

\Rightarrow 291 - 3 B D^{2} = 144

\Rightarrow 3 B D^{2} = 291 - 144

\Rightarrow 3 B D^{2} = 147

\Rightarrow B D^{2} = 49

∴ BD = 7
Hence, the given statement is true.

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