Math, Geometry

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A square and an equilateral triangle have equal perimeters. The area of the triangle is 2\sqrt {3} square inches. What is the number of inches in the length of the diagonal of the square?

Please give me a full clear explanation with the answer. Thanks!!!!!!!

  • Math, Geometry -

    the triangle of side s has altitude s√3/2
    So, it has area 1/2 (s)(s√3/2) = s^2√3/4

    So, if s^2√3/4 = 2√3,
    s = √8

    So, the triangle has perimeter 3√8

    If the square has perimeter 3√8, it has side 3√8/4 = 3/2 √2

    The diagonal of a square has length s√2, so in this case, the diagonal is 3.

  • Math, Geometry -

    Let the side of the equilateral triangle be x

    area of equilateral triangle = (1/2)x^2 sin60°
    = (1/2)x^2 (√3/2) = (√3/4)x^2

    (√3/2)x^2 = 2/√3
    3x^2 = 4
    x = 2/√3

    perimeter of triangle = 3x = 6/√3
    which is equal to the perimeter of the square, so each side of the square is 6/(4√3) = 3/(2√3)

    let the diagonal be d
    d^2 = (3/(2√3) )^2 + (3/(2√3) )^2
    = 9/12 +9/12 = 18/12 = 6/4
    d = √6/√4 = √6/2

  • arithmetic errror - go with Steve Math, Geometry -

    made a silly error ...


    (√3/2)x^2 = 2/√3
    3x^2 = 4
    x^2 = 8/3
    x = √8/√3

    I also read your area of the triangle as 2/√3
    whereas Steve took it as 2√3

  • perils of typesetting :-) -

    That's what happens when TEX gets mixed in with text!

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