To determine the area of the triangle with the given information, we start with the following given data:
- [tex]\( A = 27.8^\circ \)[/tex]
- [tex]\( B = 107.3^\circ \)[/tex]
- [tex]\( c = 4 \)[/tex] units.
Step 1: Calculate angle [tex]\( C \)[/tex]
The sum of all angles in a triangle is [tex]\( 180^\circ \)[/tex]. Therefore, we can find angle [tex]\( C \)[/tex]:
[tex]\[
C = 180^\circ - A - B = 180^\circ - 27.8^\circ - 107.3^\circ = 44.9^\circ.
\][/tex]
Step 2: Use the Law of Sines to find the area
The formula for the area of a triangle using the Law of Sines is:
[tex]\[
\text{Area} = \frac{1}{2} \cdot c^2 \cdot \frac{\sin(A) \cdot \sin(B)}{\sin(C)}
\][/tex]
First, we convert angles [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex] from degrees to radians because trigonometric functions in most calculators and mathematical functions use radians:
[tex]\[
\sin(A) = \sin(27.8^\circ) \approx \sin(0.485) \approx 0.466
\][/tex]
[tex]\[
\sin(B) = \sin(107.3^\circ) \approx \sin(1.873) \approx 0.955
\][/tex]
[tex]\[
\sin(C) = \sin(44.9^\circ) \approx \sin(0.783) \approx 0.703
\][/tex]
Substitute these sin values and side [tex]\( c = 4 \)[/tex] into the area formula:
[tex]\[
\text{Area} = \frac{1}{2} \cdot 4^2 \cdot \frac{0.466 \cdot 0.955}{0.703}
\][/tex]
[tex]\[
\text{Area} = \frac{1}{2} \cdot 16 \cdot \frac{0.44573}{0.703}
\][/tex]
[tex]\[
\text{Area} = 8 \cdot \frac{0.44573}{0.703}
\][/tex]
[tex]\[
\text{Area} \approx 8 \cdot 0.634
\][/tex]
[tex]\[
\text{Area} \approx 5.072
\][/tex]
Step 3: Compare with provided choices
The closest answer choice to our calculated approximate area (5.072) is:
[tex]\[
\approx 5 \text{ units}^2
\][/tex]
Therefore, the best answer from the choices provided is:
[tex]\[
\text{c. area} \approx 5 \text{ units}^2
\][/tex]
