Answer to Question #14838 in Trigonometry for Nathan

Question

A triangular piece of land is bounded by three straight roads, two of which intersect in a right angle. The third road makes an angle of 37.6° with one of the others. The perimeter of the land is 985 yards. What is its area, to the nearest square foot?

Answer 1

Let's denote the road triangle's sides by a, b, and c. Then we've got the following system of equations:

a + b + c = 985;
a = c* cos(37.6);
b = c* sin(37.6);

Let's plug in last two into first to solve for c:

c*cos(37.6) + c*sin(37.6) + c = 985 ==>
0.79c + 0.61c + c = 985 ==>
2.4c = 985 ==>
c = 410

now plug c into those initial equations for a and b to solve for a and b:

a = 326, b = 250: area = 1/2 * a*b = 0.5*326*250 = 40.750 square yards.