First let's use pythagorean theorem:
c² = a² + b²
Where:
a = a
c = 15
b = 8
15² = a² + 8²
225 = a² + 64
Solving for a:
Subtract 64 from both sides:
225 - 64 = a²
161 = a²
Take the square root to both sides:
√(161) = a
a = √(161)
Let's use law of sines to find θ:
8/sin(θ) = 15/sin(90)
Solving for θ:
sin(θ) = 8/15
θ = arcsin(8/15) = 32.23095264
Since the sum of the sum of the internal angles of a triangle is 180:
90 + 32.23095264 + α = 180
Solving for α:
α = 180 - 90 - 32.23095264 = 57.76904736
Using law of sines again:
8/sin(90) = h/sin(57.76904736)
Solving for h:
h = 8*sin(57.76904736) = 6.767241355