Answer:
[tex]x=6\sqrt{5}\\ \\y=3\sqrt{5}\\ \\z=6[/tex]
Step-by-step explanation:
The height of the right triangle drawn to the hypotenuse is geometric mean of two segments of hypotenuse, so
[tex]z^2=12\cdot 3\\ \\z^2=36\\ \\z=6[/tex]
By the Pythagorean theorem,
[tex]x^2=12^2+z^2\\ \\x^2=144+6^2\\ \\x^2=144+36\\ \\x^2=180\\ \\x=6\sqrt{5}[/tex]
and
[tex]y^2=z^2+3^2\\ \\y^2=6^2+9\\ \\y^2=36+9\\ \\y^2=45\\ \\y=3\sqrt{5}[/tex]
