Trigonometry quadrilateral

Question

The four sequential sides of a quadrilateral have lengths , , , and (all measured in yards). The angle between the two smallest sides is .

What is the area of this figure?

Answer 1

Using the Law of Cosines:

d² = 3.4² + 5.5² - 2(3.4)(5.5) cos 102°

d = 7.041725445

Find the area (Herron's formula) of the two triangles. One has perimeter of 3.4, 5.5 and 7.0 and the other has a perimeter of 9.7, 8.9, and 7.0

Answer 2

Assuming we have a cyclic quadrilateral, use Bretschneider's formula:

K = sqrt[(s-a)(s-b)(s-c)(s-d)-abcd*cos²((α+γ)/2)]

K is the area of the quadrilateral. s = (a+b+c+d)/2 and is the semiperimeter of the quadrilateral. In a cyclic quadrilateral, opposite angles are supplementary. Therefore, γ = 78°. Note that the cos term simplifies to 0 since cos(90°) = 0.

The simplified formula is now:

K = sqrt[(s-a)(s-b)(s-c)(s-d)]

After substituting in the given values, you should get approximately 40.95 square yards for the area.

Hope this was helpful.

EDIT: If you can't assume the quadrilateral is cyclic, then see Doug C.'s method below (credits to him for pointing this out):

1. Use the law of cosines to find the length of the diagonal used to split the quadrilateral into two triangles. The length of this diagonal is approximately 7.04 yards.
2. Then, use Heron's formula to find the area of each triangle. Area of triangle with sides a, b, and diagonal is approximately 9.11 square yards. Area of triangle with sides c, d, and diagonal is approximately 30.09 square yards.
3. Add the two areas of the triangles together to get the area of the quadrilateral, which should be approximately 39.2 square yards.