Question

What are the values the values of a and b?

Answer 1

The segment marked 20 is the geometric mean of the two pieces into which it splits the hypotenuse. You get the proportion

$\frac{20}{a} = \frac{21}{20}$

Solving,

$21a =(20)(20) \\ 21a = 400 \\ a = \frac{400}{21}$

To find  b,  use the fact that  b  is the geometric mean of the whole hypotenuse, 21 + a, and the part that is attached to it,  a.

$\frac{b}{a} = \frac{21+a}{b}$

A little arithmetic shows that $21+a = \frac{841}{21}$.

$\frac{b}{\frac{400}{21}} = \frac{\frac{841}{21}}{b} \\ b^2 = \frac{336400}{441} \\ b=\frac{580}{21}$