A cat runs due east 120 feet to a corner. After turning through an angle of [tex]\(67.8^\circ\)[/tex], the cat walks 362 feet to the second corner. Then the cat walks back to the starting point.

What is the area of the triangle formed by his path?

Round to the nearest hundredth.

[tex]\[
\text{Area} = \, \text{[?]} \, \text{ft}^2
\][/tex]

[tex]\(\square\)[/tex] Enter



Answer :

To determine the area of the triangle formed by the path of the cat, we need to follow these steps:

1. Identify the lengths of the sides of the triangle:
- The cat runs 120 feet due east.
- The cat then turns through an angle of [tex]\( 67.8^\circ \)[/tex] and walks 362 feet.
- We need to determine the length of the third side, which is the distance the cat walks back to the starting point.

2. Find the length of the third side:
To find the length of the third side, we use the Law of Cosines.

The Law of Cosines states:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(C) \][/tex]
where:
- [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the lengths of two sides of the triangle.
- [tex]\( C \)[/tex] is the included angle.
- [tex]\( c \)[/tex] is the length of the side opposite the angle [tex]\( C \)[/tex].

Here:
[tex]\[ a = 120 \, \text{feet}, \quad b = 362 \, \text{feet}, \quad C = 67.8^\circ \][/tex]

Plugging in these values:
[tex]\[ c = \sqrt{120^2 + 362^2 - 2 \times 120 \times 362 \times \cos(67.8^\circ)} \][/tex]
[tex]\[ c \approx 335.58 \, \text{feet} \][/tex]

3. Use Heron's formula to find the area of the triangle:
- Calculate the semi-perimeter [tex]\( s \)[/tex]:
[tex]\[ s = \frac{a + b + c}{2} \][/tex]
[tex]\[ s = \frac{120 + 362 + 335.58}{2} \approx 408.79 \, \text{feet} \][/tex]

- Now, apply Heron's formula, which states:
[tex]\[ \text{Area} = \sqrt{s(s - a)(s - b)(s - c)} \][/tex]

So, substituting the values:
[tex]\[ \text{Area} = \sqrt{408.79 \times (408.79 - 120) \times (408.79 - 362) \times (408.79 - 335.58)} \][/tex]

4. Calculate the area:
Plugging in the values:
[tex]\[ \text{Area} \approx \sqrt{408.79 \times 288.79 \times 46.79 \times 73.21} \approx 20109.91 \, \text{square feet} \][/tex]

Therefore, the area of the triangle formed by the cat's path is:
[tex]\[ \boxed{20109.91} \, \text{square feet} \][/tex]