geometry
posted by jay .
Kristen lives directly east of the park. The football field is directly south of the park. The library sits on the line formed between Kristenâ€™s home and the football field at the exact point where an altitude to the right triangle formed by her home, the park, and the football field could be drawn. The library is 2 miles from her home. The football field is 8 miles from the library.
How far is library from the park?
How far is the park from the football field?
A. 4 miles; miles
B. miles; miles
C. 5 miles; 9 miles
D. 4 miles; miles

Trying to match the question with the choices sure stumps me.
Using similar triangles, if we label the vertices of the triangle P,K,F, and the library L, then we have
∆LPF ~ ∆KLP
we want the distance LP and FP, and we have
LP/2 = FP/KP
LP/8 = KP/FP
so,
LP/2 = 8/LP
LP^2 = 16
LP = 4
FP^2 = LP^2 + LF^2 = 4^2+8^2 = 80
FP = 4√5