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If A be the area of a right triangle and b is one of the sides containing the right angle. Find m if the length of the altitude on the hypotenuse is $\frac{mAb}{\sqrt{b^{4} + 4 A^{2}}}$ .

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answer is A.

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Detailed Solution

Given,
Right triangle area = A
One side containing Right angle = b
The hypotenuse =

\frac{mAb}{\sqrt{b^{4} + 4 A^{2}}}

We know that area of triangle

A = \frac{1}{2} \times (b) \times (h)

Now, suppose that
Another side of the triangle = x
So, area of triangle,

A = \frac{1}{2} \times b \times x

\Rightarrow x = \frac{2 A}{b}

By using the concept of Pythagoras theorem, third or hypotenuse side

= \sqrt[2]{b^{2} + \frac{4 A^{2}}{b^{2}}}

[put the value hypotenuse

= \sqrt[2]{b^{2} + x^{2}}

]

= \frac{\sqrt[2]{b^{4} + 4 A^{2}}}{b}

Length of altitude on the hypotenuse given as,

= \frac{m \times A \times b}{\sqrt[2]{b^{4} + 4 A^{2}}}

Finally, we have with all the value we calculate area of the triangle

A = \frac{1}{2}

x (hypotenuse) x (length of altitude on hypotenuse)

A = \frac{\sqrt[2]{b^{4} + 4 A^{2}} \times m \times A \times b}{\sqrt[2]{b^{4} + 4 A^{2}} \times 2 b}

\Rightarrow m = 2

Hence the required value of

m = 2

.
Correct option is 1.

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