Question

The edges of a plot of land are bordered on all sides by a forest forming a triangular plot. The edges of the plot of land are 35 m, 52 m, and 82 m. What are the angles of the border of the plot of land? Select all that apply.
(2 points)
Responses

140°
140°

74°
74°

66°
66°

40°
40°

24°
24°

16°

Answer 1

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:
- \(140°\)
- \(24°\)
- \(16°\)