I'll attempt this problem in a different way ...
Draw right triangle(ABC) with C the right angle.
Let AC = 8 and BC = 15.
Using the Pythagorean Theorem AB = 17.
Drop a perpendicular from angle(C) to the hypotenuse AB.
Label the point of intersection P.
PC is the height.
By similar triangles (or a theorem from your textbook):
PA / AC = AC / AB ---> PA / 8 = 8 / 17 ---> PA = 64 / 17
PB / BC = BC / BA ---> PB /15 = 15 / 17 ---> PB = 225 / 17
Continuing with similar trianges (or another theorem):
AP / PC = PC / PB ---> ( 64 / 17 ) / PC = PC / ( 225 / 17 )
cross-multiplying: ( 64 / 17 ) · ( 225 / 17 ) = PC2
( 64 · 225 ) / ( 17 · 17 ) = PC2
14,400 / 289 = PC2
120 / 17 = PC
as a decimal: 7.0588...