Let O be the point on leg GK directly below the vertex at H.
Then triangles GOH and HOK are similar triangles, so that corresponding sides occur in a fixed ratio with one another. In particular,
GO/H.O = H.O/KO
5/x = x/15
x² = 75
x = √75
x = √(3 • 25)
x = 5√3
Solve for y and z using the Pythagorean theorem.
x² + 15² = y²
75 + 225 = y²
y = √300
y = √(3 • 100)
y = 10√3
x² + 5² = z²
75 + 25 = z²
z = √100
z = 10
