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The area of a rectangle is 24, and its diagonal is radical 52. Find its dimensions and perimeter.

Longer side = ?
Shorter side = ?
Perimeter = ?

  • math -

    Let the dimensions be a and b.
    ab = 24
    a^2 + b^2 = 52

    a^2 + (24/a)^2 = 52

    a^4 -52a^2 + 576 = 0
    (a^2 -36)(a^2 -16)
    a^2 = 36 or 16
    a = 6 or 4
    b = 4 or 6

  • math -

    Thank You!

  • math -

    Let X and Y represent the sides of
    the rectangle.

    X^2 + Y^2 = 52 = diagonal squared.
    XY = 24 =the area of rectangle.
    y = 24/X
    X^2 + (24/X)^2 = 52
    X^2 +576/X^2 = 52
    Multiply both sides by x^2 :
    X^4 + 576 = 52X^2
    X^4 - 52X^2 + 576 = 0. used computer to
    help find roots: X = -4, X = 4, X = -6,
    X = 6. Select the positive roots:
    The sides are 4 and 6. P = (2*4) +(2*6)
    = 20

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