Question

How do you solve the triangle if it has hypotenuse 18, the opposite angle is the height, and the adjacent meets the hypotenuse at a 35 degree angle?

Answer 1

Assuming you meant "the opposite side is the height"

If we label the triangle `ABC` with
`/_A = 35^@`
and
`/_B=90^@`

then
`/_C = 55^@`

Given `c=18` (`c` being the side opposite `/_C`)

then we can use the Law of Sines:
`color(white)("XXXX")` `(sin C)/c = (sin A)/a = (sin B)/b`
and since our only missing values are `a` and `b`
the results can easily be calculated (with a calculator or with spreadsheet operations)

Answer 2

C = 90; B = 35; and A = 90 - 35 = 55 deg; hypotenuse = 18

Side CB = AB.cos 35 = 18.cos 35 = 18(0.82) = 14.75

The height, or side AC = 18.sin 35 = 18(0.57) = 10.32

Check: a^2 = b^2 + c^2

324 = 217.56 + 106.50 = 324.06 . OK