If in an oblique △ABC △ 𝐴 𝐵 𝐶 , side a=17 𝑎 = 17 ft., side b=20 𝑏 = 20 ft., and ∠C=19 ∠ 𝐶 = 19 degrees, then what is the length of side c to the nearest foot?(1 point) Responses 7 ft. 7 ft. 361 ft. 361 ft. 46 ft. 46 ft. 19 ft.

Answer 1

To find the length of side \( c \) in triangle \( ABC \), we can use the Law of Cosines, which is given by the formula:

\[ c^2 = a^2 + b^2 - 2ab \cos(C) \]

where:

• \( a = 17 , \text{ft} \)
• \( b = 20 , \text{ft} \)
• \( C = 19^\circ \)

Now we will substitute the values into the formula:

1. Calculate \( a^2 \) and \( b^2 \):

\[ a^2 = 17^2 = 289 \] \[ b^2 = 20^2 = 400 \]

2. Calculate \( 2ab \):

\[ 2ab = 2 \cdot 17 \cdot 20 = 680 \]

3. Calculate \( \cos(19^\circ) \):

Using a calculator, we find:

\[ \cos(19^\circ) \approx 0.9455 \]

4. Now substitute all these into the Law of Cosines formula:

\[ c^2 = 289 + 400 - 680 \cdot 0.9455 \]

5. Calculate \( 680 \cdot 0.9455 \):

\[ 680 \cdot 0.9455 \approx 642.34 \]

6. Substitute this back into the formula:

\[ c^2 = 289 + 400 - 642.34 \] \[ c^2 = 689 - 642.34 \approx 46.66 \]

7. Now take the square root of both sides to find \( c \):

\[ c \approx \sqrt{46.66} \approx 6.83 \]

Rounding this to the nearest foot, we find:

\[ c \approx 7 , \text{ft} \]

Thus, the length of side \( c \) to the nearest foot is \( \boxed{7 , \text{ft}} \).