Carpenter's Question

The section of valley $\overline{AJ}$ has to rise the same amount for right and left sections of the roof. If it goes up 4 inches on the right, as it does in $\triangle AFG$ , it has to go up 4 inches in $\triangle ADE$ . The right $\triangle ABC$ , showing the pitch on the left, goes in horizontally 12 inches, but goes up only 3 inches. To get the required 4 inches, we need to go farther in. Triangles $\triangle ABC$ and $\triangle ADE$ are similar, so the required horizontal distance is 4/3 of 12, or 16 inches.

Triangle $\triangle AGJ$ , significantly enlarged for clarity, is shown above against the piece of plywood. It is similar to $\triangle KLJ$ , so

$\frac{x}{4}=\frac{16}{y}$

$y$ can be obtained from the $\triangle AFG$ (previous figure) using Pythagorean theorem. $y=\sqrt{12^2+4^2}=12.649$ . Therefore $x=4\times\frac{16}{12.649}=5.0596$ feet, which is approximately 60 and 3/4 of an inch.