Final answer:
To find the dimensions of the door, we set up equations based on the given information that the rod, which measures as the door's diagonal, is 4 ch'ih longer than the door's width and 2 ch'ih longer than its height. Using these relationships and the Pythagorean theorem, we can determine the width and height by solving a quadratic equation.
Step-by-step explanation:
To solve this problem, let's designate the width of the door as w ch'ih and the height as h ch'ih. Given that the rod is also the diagonal of the door and is 4 ch'ih longer than the width and 2 ch'ih longer than the height, it means we can set up the following relationships: the length of the rod as w + 4 and h + 2.
Since the rod equals the diagonal, by the Pythagorean theorem, we can say that (w + 4)^2 = w^2 + h^2.
We are also given that the rod is 2 ch'ih longer than the height, which means h + 2 = w + 4. Simplifying this gives us h = w + 2. Substituting into the Pythagorean theorem, we get (w + 4)^2 = w^2 + (w + 2)^2.
Expanding the squares and simplifying, we end up with a quadratic equation in terms of w. Solving for w will give us the width of the door, and through substitution, we can then find the height.