question_answer

If A is the area of a right angled triangle and b is one of the sides containing the right angle, then what the length of the altitude on the hypotenuse?

A)  \[\frac{2\,\,Ab}{\sqrt{{{b}^{4}}+4{{A}^{2}}}}\]      

B)  \[\frac{2\,\,{{A}^{2}}b}{\sqrt{{{b}^{2}}+4{{A}^{2}}}}\]

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

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

Correct Answer: A

Solution :

In \[\Delta ABC,\]

\[A=\frac{1}{2}\times \]base \[\times \]altitude \[=\frac{1}{2}\times b\times AC\]

\[AC=\frac{2A}{b}\]

Using Pythagorus theorem,

\[A{{C}^{2}}+A{{B}^{2}}=B{{C}^{2}}\]

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

Again in \[\Delta ABC,\]\[A=\frac{1}{2}\times BC\times AD\]

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