Question

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

Answer 1

To find the length of side \( c \) in triangle \( ABC \) where side \( a = 17 \) ft, side \( b = 20 \) ft, and angle \( C = 19^\circ \), we can use the Law of Cosines. The formula is:

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

Plugging in the given values:

\[
c^2 = 17^2 + 20^2 - 2 \cdot 17 \cdot 20 \cdot \cos(19^\circ)
\]

Calculating each part:

1. \( 17^2 = 289 \)
2. \( 20^2 = 400 \)
3. \( \cos(19^\circ) \approx 0.9455 \)

Now substitute these into the equation:

\[
c^2 = 289 + 400 - 2 \cdot 17 \cdot 20 \cdot 0.9455
\]

Calculating \( 2 \cdot 17 \cdot 20 \):

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

Now substitute this back into the equation:

\[
c^2 = 689 - 680 \cdot 0.9455
\]

Calculating \( 680 \cdot 0.9455 \):

\[
680 \cdot 0.9455 \approx 642.34
\]

Now, substitute this back:

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

To find \( c \):

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

Rounding to the nearest foot, the length of side \( c \) is approximately \( 7 \) ft.

Thus, the final answer is:

**7 ft**.