Arectangular painting has a diagonal measure of 5 inches and an area of 12 square inches. use the formula for the area of a rectangle and the pythagorean theorem to find the length and width of the painting.

We have 2 different equations with which we need to work to find the answer to this. First off, the area formula is A = L * W. Our area is 12, so 12 = L * W. Then we have the Pythagorean's theorem we have to worry about, which says that . We are going to solve the area equation for L to get . What we are going to do now is sub that into the other formula in place of L to get and . In order to solve for w, we need to multiply all those terms by w^2 to get rid of the denominator. That leaves us with this then: . We need to factor that now to solve for w. We will bring everything over to the one side of the equation and set it equal to 0. . We are going to use a u substitution here to make the factoring easier. We are going to let so . Now we will factor . When we factor that we get (u - 16)(u - 9). By the Zero Product Property, either u - 16 = 0 or u - 9 = 0. u = 16 and u = 9. But remember that u = w^2, so what we really have, after we make the replacement, is so w = 4, and and w = 3. Now it just so happens that if the length is 4 and the width is 3, the product of those is 12 (the area), and if you remember, in a right triangle with a hypotenuse of 5, the side lengths will be 3 and 4. So your dimensions of the rectangle are that the length is 4 and the width is 3. There you go!

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