Answer:
13.1 square units
Step-by-step explanation:
To calculate the area of the triangle given the lengths of all three sides and the measure of one angle, we can use the Area Sine Rule:
[tex]\boxed{\begin{array}{l}\underline{\text{Area Sine Rule}}\\\\A=\dfrac{1}{2}ab \sin C\\\\\text{where:}\\ \phantom{ww}\bullet \;\text{$C$ is the angle.} \\ \phantom{ww}\bullet \;\text{$a$ and $b$ are the sides enclosing the angle.}\end{array}}[/tex]
In this case:
Substitute the values into the formula and solve for area A:
[tex]A=\dfrac{1}{2} \cdot 4.6 \cdot 7.7 \cdot \sin 132.2^{\circ} \\\\ A=17.71 \sin 132.2^{\circ} \\\\A=13.119649400238... \\\\A = 13.1\; \sf square\;units\;(1\;d.p.)[/tex]
Therefore, the area of the given triangle rounded to one decimal place is:
[tex]\LARGE\boxed{\boxed{13.1\; \sf square\;units}}[/tex]