Question

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

Answer 1

In $△ABC$, by Pythagoras theorem

$B{C}^2=A{B}^2+C{A}^2$

$B{C}^2=c^2+b^2$

In $△ABD$ and $△CBA$

$\angle ABD=\angle CBA$

$\angle ADB=\angle CAB$

So by AA, $△ABD\sim △CBA$

As corresponding parts of similar triangles are equal so $\frac{AB}{CB}=\frac{AD}{CA}$

$\frac{c}{c^2+b^2}=\frac{AD}{b}$

$AD=\frac{cb}{c^2+b^2}$