Here's the general idea.
(Note: I haven't taken physics; I'm more of a math guy, so please excuse any improper notation or conventions I use. =D) There are three forces acting on the weight - gravity, the pulling from the segment of string attached to the first peg (call it peg A), and the pulling from the segment of string attached to the other peg (peg B). Call these forces F_g, F_a, and F_b, respectively. We want to find all these forces, because F_a is equal and opposite to the force the string applies on peg A, which is what you're looking for.
F_g, F_a, and F_b must sum to zero, because the weight isn't moving. However, we can't solve for F_a and F_b, because we don't know what direction the forces are applied in. We need to know the angles of each segment of string.
Note that since the string is stretched straight, the two pegs and the weight (more specifically, the point where the weight attaches to the string) form a triangle. You can find all the side lengths of the triangle, so you can solve the triangle and find the angles of the two strings using trigonometry.
Once you find the angles, just split the forces into components, sum to 0, and solve.
