A farmer has a triangular field with sides that measure 120 yards, 170 yards, and 220 yards. Find the area of the field
call the vertices A,B,C
Use law of cosines to find A.
area = 1/2 * AB * AC*sinA
To find the area of the triangular field, you can use Heron's formula. Heron's formula allows you to calculate the area of a triangle when you know the lengths of all three sides. Here's how you can use it to find the area of the triangular field:
Step 1: Calculate the semi-perimeter (s) of the triangle. The semi-perimeter is half the sum of the lengths of all three sides. In this case, the lengths of the sides are 120 yards, 170 yards, and 220 yards. So, the semi-perimeter is:
s = (120 + 170 + 220) / 2
= 510 / 2
= 255 yards
Step 2: Plug the semi-perimeter value into Heron's formula. Heron's formula for the area (A) of a triangle is given by:
A = sqrt(s * (s - a) * (s - b) * (s - c))
Where 'a', 'b', and 'c' are the lengths of the sides of the triangle. In our case, using the lengths of the sides (120, 170, and 220) and the semi-perimeter value (255), we can calculate the area:
A = sqrt(255 * (255 - 120) * (255 - 170) * (255 - 220))
Step 3: Evaluate the expression to find the area:
A ≈ sqrt(255 * 135 * 85 * 35)
≈ sqrt(432562875)
≈ 20799.037
So, the area of the triangular field is approximately 20799.037 square yards.