Kelly, even if 3 and 11.5 are a straight line, there is still no possible way to solve x unless you assume a right angle or a parallel line somewhere. You MUST have 3 pieces of info of a triangle, at least one of them being a side, which we do not have. Being symmetrical here does not help.
The only assumptions I made were those I stated.
Here's some more info:
Extend the dotted line, and let's just consider the geometry to the left (since it's symmetric). Basic algebra tells us that if we can write 'n' independent equations, with 'n' unknowns, then we should be able to solve for all variables. So here are some equations we can write:
a + b = 17 (half of the 34 dimension)
b^2 + d^2 = 11.5^2 (Pythagorean theorem on the right triangle we formed by extending x) (This is a right triangle, assuming the drawing is symmetric, as the line 'x' will bisect the bottom angle and this is an isosceles triangle)
Let's call the vertical angles created where the 3/11.5 line crosses the horizontal line angle y. Now we can write
sin(y) = d/11.5 (definition of sin) and
7.5^2 = 3^2 + a^2 - 2*3*a*cos(y) (Law of cosines, applied to the triangle on the left).
We now have 4 independent equations, and 4 unknowns (a, b, d, y). Now we can use algebra, trial and error, recursive calculations, or whatever to solve and find all of these. You can also see that ' d + x' can be found from the Pythagorean theorem (17^2 + (d+x)^2 = 45^2), so once you have d, it is easy to get x.
There's many ways to do this - the key is seeing that the upper left triangle and the 'new' triangle I formed (or, the middle triangle) have an angle in common, and that locks down the geometry, and allows us to write as many equations as we have unknowns.
