Question

Using Pythagoras theorem determine the length of AD in terms of b and c shown in Fig.

Answer 1

We have, 

In ∆BAC, by Pythagoras theorem, we have 

BC² = AB² + AC² 

⇒ BC² = c² + b² 

⇒ BC = √(c² + b²) 

In ∆ABD and ∆CBA 

∠B = ∠B  [Common] 

∠ADB = ∠BAC  [Each 90°] 

Then, ∆ABD ͏~ ∆CBA   [By AA similarity] 

Thus, \(\frac{AB}{CB} = \frac{AD}{CA}\) [Corresponding parts of similar triangles are proportional] 

\(\frac{c}{\sqrt{(c^2 + b^2)}}\) = \(\frac{AD}{b }\)

∴ AD = \(\frac{bc}{\sqrt{(c^2 + b^2)}}\)