Geometry

In quadrilateral ABCD, we have AB = BC = CD = DA, AC = 14, and BD = 48. Find the perimeter of ABCD.

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  1. A quadrilateral with all four sides equal is a rhombus, whose area is half the product of the diagonals AC and BD.

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    posted by MathMate
  2. Yeah, but we're not trying to find the area?

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  3. Sorry.
    In that case, we make use of the fact that rhombuses have diagonals which intersect at right-angles.
    Thus each side is the hypotenuse of a right triangle with legs equal to the half-diagonals.

    Thus if s=length of each side of the rhombus, then
    s²=(AC/2)²+(BD/2)²
    using Pythagoras theorem.
    Calculate
    s=√(7²+24²)
    =25

    The perimeter is 4s.

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    posted by MathMate
  4. Stop cheating Sheenybeany!

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    posted by AoPS

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