The perimeter of a rectangle is 68 inches it diagonal measure 26 inches what are the masure of the lenght and with of the rectangle?

Answer 1

The diagonal of a rectangle divides the rectangle into two right triangles of equal area, with the length of the diagonal as the hypotenuse. The perimeter is twice the sum of the length and the width. Designating the length and width by l and w respectively, one obtains two equations: 2w + 2l = 68 and, from the Pythagorean theorem, w2 + l2 = 262. From the first of these equations, l = 34 -w. Substituting this relationship into the second equation yields w2 + (34 - w)2 = 676. Multiplying out the binomial square yields w2 + 342 - 68w + w2 = 676. Dividing by 2, adding the coefficients of w2, and reducing to standard form yields w2 -34w + 240 = 0. This can be factored into (w - 24)(w - 10) = 0, which can be true either for w = 24 or w = 10. The correct one of these can be chosen by noting that, from the definitions of length and width, l > w, which is not true for w = 24 -- if w were 24, the perimeter would be at least 4 X 24 = 96, which is inconsistent with the stated perimeter. Therefore, w = 10 and l = 24.