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Sum Adjacent Lengths (Posted on 2007-10-21) Difficulty: 2 of 5
The two diagonals of a cyclic quadrilateral EFGH are EG and FH while
the respective lengths of its adjacent sides EF and FG are 2 and 5.

It is known that Angle EFG = 60o and the area of the quadrilateral is 4√3.

Determine the sum of the lengths GH + HE.

See The Solution Submitted by K Sengupta    
Rating: 2.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution Solution Comment 1 of 1

Let e = |GH| and g = |HE|.
Area = 4*sqrt(3)
     = (1/2)|EF||FE|sin(60) + (1/2)eg*sin(180-60)
    or
  eg = 6
|EG| = |EF|^2 + |FG|^2 - 2|EF||FG|cos(60)
     = e^2 + g^2 - 2eg*cos(180-60)
    or
  19 = e^2 + g^2 + eg
    or
  25 = 19 + eg = (e+g)^2
    or
  |GH|+|HE| = 5
One of the two possible quadrilaterals is an
Isosceles Trapezoid.
 

  Posted by Bractals on 2007-10-21 14:02:43
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