The segment marked 20 is the geometric mean of the two pieces into which it splits the hypotenuse. You get the proportion
[tex]\frac{20}{a} = \frac{21}{20}[/tex]
Solving,
[tex]21a =(20)(20) \\ 21a = 400 \\ a = \frac{400}{21} [/tex]
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.
[tex]\frac{b}{a} = \frac{21+a}{b}[/tex]
A little arithmetic shows that [tex]21+a = \frac{841}{21}[/tex].
[tex]\frac{b}{\frac{400}{21}} = \frac{\frac{841}{21}}{b} \\ b^2 = \frac{336400}{441} \\ b=\frac{580}{21} [/tex]
