Question

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

(a) \(\frac{2Ab}{\sqrt{b^4+4A^2}}\)

(b) \(\frac{2A^2b}{\sqrt{b^4+4A^2}}\)

(c) \(\frac{2Ab^2}{\sqrt{b^4+4A^2}}\)

(d) \(\frac{2A^2b^2}{\sqrt{b^4+A^2}}\)

Answer 1

(a) \(\frac{2Ab}{\sqrt{4A^2+b^4}}\)

Let ABC be the given right angled triangle, right angled at B. Let BC = b. 

Then, Area of ΔABC = \(\frac12\) x BC x AB 

⇒ A = \(\frac12\) x b x AB ⇒ AB = \(\frac{2A}{b}\)

In ΔABC, AC² = AB² + BC²

AC² = \(\frac{4A^2}{b^2}+b^2\)

^{⇒ AC =} \(\sqrt{\frac{4A^2}{b^2}+b^2}\)

Again in ΔABC,

 A = \(\frac12\) x AC x BD

⇒ A = \(\frac12\) x \(\sqrt{\frac{4A^2}{b^2}+b^2}\) x BD

⇒ BD = \(\frac{2Ab}{\sqrt{4A^2+b^4}}.\)