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a rod of unknown length is used to measure the dimensions of a rectangular door. the rod is 4 ch'ih longer than the width of the door. the rod is 2 ch'ih longer than the height of the door, ad the same length as the door's diagonal. what are the dimensions of the door?

User Hgcummings
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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.

User Keva
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