Here's how I got 13.98:
You want to figure out the length of the unknown segment of the triangle (X)
Draw a line C bisecting the 120 degree angle that's perpendicular to the unknown side X , splitting the unknown side into two segments (Let's call them D and E) and the 120 degree angle into two angles (let's call them F and G) .
Now you have two right triangles, one comprised of sides C and D with hypotenuse of 4.2"; the other comprised of sides C and E = X - D with a 11.4" hypotenuse
Solve for C and D by noting that the right triangle with the 4.2" hypotenuse has two angles of 45 degrees (B, which is 180 - 120 - 15, and F, which is 180 - 90 - B).
- Therefore, the two sides have to be equal length, and a^2 + b^2 = c^2 can be simplified to 2a^2 = c^2
- 2a^2 = (4.2)^2 ---> a = 2.97 = C = D
Solve for E:
- a^2 + b^2 = c^2
- C^2 + E^2 = (11.4)^2
- (2.97)^2 + E^2 = (11.4)^2 ---> E = 11.01
finally, segment X = D + E = 2.97 + 11.01 = 13.98