Dividing a quarilateral

[miamivince]

A quad with side lengths 34 , 54 , 58 , and 78 meters is divided by a diagonal that gives 2 areas, 1 of which is 50 % greater than the other. Find the length of the diagonal d ?
Looks easy, I found the area of the quad by Area sqrd = (s-b)(s-c)(s-e)(s-g) formula. However this wasn't a perfect square. The sides 54 and 58 are supposed to be opposite one another. I assume the 2 new shapes are a smaller quad and a triangle . The answer without workings is given as 36. Thanks.

 

[DrPhil]

A quad with side lengths 34 , 54 , 58 , and 78 meters is divided by a diagonal that gives 2 areas, 1 of which is 50 % greater than the other. Find the length of the diagonal d ?
Looks easy, I found the area of the quad by Area sqrd = (s-b)(s-c)(s-e)(s-g) formula. However this wasn't a perfect square. The sides 54 and 58 are supposed to be opposite one another. I assume the 2 new shapes are a smaller quad and a triangle . The answer without workings is given as 36. Thanks.

Knowing the four sides of a quadrilateral is NOT sufficient for finding its area. Also, any diagonal divides the shape into two triangles

There are at least six cases to consider (unless you are given more than you told us!). Were you Given that sides 54 and 58 are opposite each other? Do you know that the quadrilateral is convex? That would reduce the number of cases to two!

1) the diagonal forms triangles with sides 34-54-d, and with sides 58-78-d
2) the diagonal forms triangles with sides 34-58-d, and with sides 54-78-d

Hints: Each case has limits for min and max d by rules of triangles, but d is a variable.
........The area of the triangle including 78 is 1.5 times the area of the triangle with 34.

Let us see your work!