A farmer owns a triangular piece of land. He wants to find the
length of the back of his property, which is mostly bog land. He
knows that his left and right boundary lines connect near his house at
an angle of 143°. The left boundary length is 95 m and the right is 110
m. Help the farmer determine the length of his rear property line
The farmer can use the Law of Cosines to find the length of the rear property line (c). The Law of Cosines for a triangle with sides of lengths a, b, and c, and an included angle θ, is as follows:
c = sqrt(a^2 + b^2 - 2*a*b*cos(θ))
Here, a=95 m, b=110 m, and θ=143 degrees.
First, convert the angle to radians, because the cosine function in most programming and calculation tools uses radians, not degrees. Remember that π radians = 180 degrees, so θ = 143*(π/180)
Then plug the values into the formula:
c = sqrt((95^2) + (110^2) - 2*95*110*cos(143*π/180))
After performing the calculation, the length of the rear property line (c) will be approximately 189.94 meters.
To determine the length of the rear property line, we can use the Law of Cosines.
The Law of Cosines states that for any triangle with sides a, b, and c, and angle C opposite side c, the following equation holds:
c^2 = a^2 + b^2 - 2ab*cos(C)
In this case, our triangle has sides a = 95 m, b = 110 m, and angle C = 143°.
First, we need to convert the angle from degrees to radians:
C_radians = C * (π/180)
C_radians = 143 * (π/180)
C_radians ≈ 2.494 m
Now, we can substitute the values into the formula and solve for c:
c^2 = 95^2 + 110^2 - 2(95)(110)*cos(2.494)
Simplifying the equation:
c^2 ≈ 9025 + 12100 - 2(95)(110)*cos(2.494)
c^2 ≈ 21125 - 20958.6*cos(2.494)
c^2 ≈ 21125 - 20958.6*(-0.791605153)
c^2 ≈ 21125 - (-16569.09574)
c^2 ≈ 37694.09574
c ≈ √(37694.09574)
c ≈ 194.16 m
Therefore, the length of the rear property line is approximately 194.16 m.