The area of a rectangle is 16, and its diagonal is 68−−√. find its dimensions and perimeter.

The perimeter of a rectangle is the sum of twice the length and twice the width.  When a diagonal is drawn across the rectangle, two congruent triangles are formed.  The legs of the triangle are the sides of the rectangle.  Right triangles indicates that we can use the Pythagorean theorem. Let length = xLet width = y  Set the equation based on what we know. 2x + 2y = 22 x + y = 11                   eq1  x2 + y2 = [√65]2               eq2   We have two equations to work with. x + y = 11                eq1 x2 + y2 = 65               eq2  Substitute eq1 into eq2.  From eq1, x = 11 - y x2 = (11 - y)2 x2 = (11 - y)(11 - y) x2 = 121 - 22y + y2  (121 - 22y + y2) + y2 = 65 121 - 22y + 2y2 = 65 2y2 - 22y + 56 = 0 2(y2 - 11y + 28) = 0 2(y - 7)(y - 4) = 0 y = 7       and        y = 4  Substitute these y values into eq1 to solve for x. x = 11 - 7                and              x = 11 - 4 x = 4                       and              x = 7  Each pair of solutions has the same dimensions.  Shorter side = 4Longer side = 7

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