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geometry

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The longer diagonal of a parallelogram measures 62 cm and makes an angle of 30 degrees with the base. Find the area of the parallelogram if the diagonals intersect at angle of 70 degrees.

  • solid mensuration -

    1405

  • geometry -

    Since the angle of intersection of the diagonals is 70 degrees then the other angle of intersection of the diagonals is 110 degrees.

    180 = 70 + θ
    where θ = the other angle of intersection
    θ = 110 degrees

    A parallelogram makes 4 oblique triangles. One triangle has angles : 30, 40, and 110 degrees.
    Since the diagonals intersect at their midpoints, the length of a side of one oblique triangle will measure 31 degrees (which is from 62/2).
    By the given values you can solve for half the length of the shorter diagonal using the law of sines.
    31/sin40 = x/sin30
    where x = the length of one side of the oblique triangle and also half the length of the shorter diagonal.
    x = 24.11371932 cm
    2(24.11371932) = 48.23 --> length of shorter diagonal

    use the formula:
    Area of parallelogram = (lenth of longer diagonal x lenth of shorter diagonal x sinθ)/2
    where θ = angle of intersection of the two diagonals (either 70 degrees or 110 degrees)

    A = ((62 cm)(48.23)sin70)/2
    A = 1404.962628 or 1405 cm^2

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