+21 Let $\overline{AB}$ be a diameter of a circle, and let $C$ be a point on the circle such that $AC = 8$ and $BC = 4.$ The angle bisector of $\angle ACB$ intersects the circle at point $M.$ Find $CM.$
[asy]
unitsize(0.5 cm);
pair A, B, C, D, P;
A = (4,0);
B = (0,3);
C = (0,0);
D = (12/7,12/7);
P = intersectionpoint(D--(D + 3*(D - C)), circumcircle(A,B,C));
draw(A--B--C--cycle);
draw(circumcircle(A,B,C));
draw(C--P);
label("$A$", A, SE);
label("$B$", B, NW);
label("$C$", C, SW);
dot("$M$", P, NE);
[/asy]
First, we can use the Pythagorean theorem to find the length of AC. Since AB is a diameter of the circle, we have:
AC^2 = 4^2 + 8^2 = 80
AC = 4*sqrt(5)
Now, let's draw the angle bisector of ∠ACB and label the point where it intersects the circle as M. We can use the angle bisector theorem to find the ratio of AM to MC:
AM/MC = AB/BC AM/MC = 8/4 AM/MC = 2
We also know that AM + MC = AC, so we can substitute in the values we found for AM/MC and AC:
2MC + MC = 4*sqrt(5)
3MC = 4*sqrt(5)
MC = (4/3)*sqrt(5)
Therefore, CM = (4/3)*sqrt(5).
