[asy] pair A=(0,0), B=(12,0), C=(0,17), Y=(10,0), Z=(0,8); draw(A--B--C--cycle); draw(B--Y); draw(C--Z); draw(Y--Z); label("A", A, SW); label("B", B, SE); label("C", C, N); label("Y", Y, SE); label("Z", Z, NW); [/asy]
By the Angle Bisector Theorem, we have:
\frac{AY}{BZ} = \frac{CY}{CZ}
Substituting in the given values, we get:
\frac{10}{BZ} = \frac{8}{CZ}
Cross-multiplying, we get:
10CZ = 8BZ
Also, by the Pythagorean Theorem on triangle BYZ, we have:
BZ^2 + BY^2 = BZ^2 + (CZ - CY)^2
Since BY is the angle bisector of angle ABC, we know that triangle ABC is isosceles, so AB = AC = 12. Therefore, BY = AY + YC = 10 + 8 = 18. Substituting this into the Pythagorean Theorem, we get:
BZ^2 + 18^2 = BZ^2 + (CZ - 8)^2
Subtracting BZ^2 from both sides, we get:
18^2 = (CZ - 8)^2
Taking the square root of both sides, we get:
CZ - 8 = 18
Adding 8 to both sides, we get:
CZ = 26
Substituting this into the equation 10CZ = 8BZ, we get:
10(26) = 8BZ
Dividing both sides by 8, we get:
BZ = \frac{10(26)}{8} = 32.5
Therefore, BZ = 32.5 and BC = BZ + CZ = 32.5 + 26 = 58.5.
Answer: BZ = 32.5, BC = 58.5
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