To find the angles of the triangular plot of land with sides 35 m, 52 m, and 82 m, we can use the Law of Cosines. This law states that for a triangle with sides \(a\), \(b\), and \(c\), and the angles opposite those sides as \(A\), \(B\), and \(C\), respectively:
\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]
First, let's assign the sides to the variables:
- \(a = 35\) m
- \(b = 52\) m
- \(c = 82\) m
We can find the angles using the Law of Cosines.
Step 1: Calculate Angle C (opposite side c)
Using the formula:
\[ 82^2 = 35^2 + 52^2 - 2(35)(52) \cos(C) \]
Calculating each term:
- \(82^2 = 6724\)
- \(35^2 = 1225\)
- \(52^2 = 2704\)
Substituting into the equation:
\[ 6724 = 1225 + 2704 - 2(35)(52) \cos(C) \]
Calculating \(1225 + 2704 = 3929\). Thus,
\[ 6724 = 3929 - 3640 \cos(C) \]
Rearranging gives:
\[ 3640 \cos(C) = 3929 - 6724 = -2795 \] \[ \cos(C) = -\frac{2795}{3640} \approx -0.7685 \]
Now, finding angle \(C\):
\[ C = \cos^{-1}(-0.7685) \approx 140° \]
Step 2: Calculate Angle A (opposite side a)
Using the formula:
\[ 35^2 = 52^2 + 82^2 - 2(52)(82) \cos(A) \]
Calculating:
- \(35^2 = 1225\)
- \(52^2 = 2704\)
- \(82^2 = 6724\)
Substituting into the equation:
\[ 1225 = 2704 + 6724 - 2(52)(82) \cos(A) \]
Calculating \(2704 + 6724 = 9428\). Thus,
\[ 1225 = 9428 - 8528 \cos(A) \]
Rearranging gives:
\[ 8528 \cos(A) = 9428 - 1225 = 8203 \] \[ \cos(A) = \frac{8203}{8528} \approx 0.9631 \]
Now, finding angle \(A\):
\[ A = \cos^{-1}(0.9631) \approx 16° \]
Step 3: Calculate Angle B (using the angles sum property)
Since the sum of angles in a triangle is \(180°\), we can find angle \(B\):
\[ B = 180° - A - C = 180° - 16° - 140° = 24° \]
Summary of Angles
The angles of the triangular plot of land are approximately:
- Angle A: \(16°\)
- Angle C: \(140°\)
- Angle B: \(24°\)
Final Answer
The angles of the border of the plot of land among the options are:
- \(140°\)
- \(24°\)
- \(16°\)