Answer:
[tex]A=34\ units^2[/tex]
Step-by-step explanation:
Suppose we have a general triangle like the one shown in the figure.
We know the angle A, the angle B and the length b.
[tex]A = 30\°\\\\B = 45\°\\\\b = 10[/tex]
By definition I know that the sum of the internal angles of a triangle is always equal to 180 °.
So
[tex]A + B + C = 180\\\\30 + 45 + C = 180[/tex]
We solve the equation and thus we find the angle C.
[tex]C = 180 - 30-45\\\\C = 105[/tex]
We already know the three triangle angles.
Now we use the sine theorem to calculate the sides c and a.
The sine theorem says that:
[tex]\frac{sin(A)}{a}=\frac{sin(B)}{b}=\frac{sin(C)}{c}[/tex]
Then
[tex]\frac{sin(30)}{a}=\frac{sin(45)}{10}[/tex]
[tex]\frac{sin(30)}{\frac{sin(45)}{10}}=a[/tex]
[tex]a=7.071[/tex]
Also
[tex]\frac{sin(105)}{c}=\frac{sin(45)}{10}[/tex]
[tex]\frac{sin(105)}{\frac{sin(45)}{10}}=c[/tex]
[tex]c=13.660[/tex]
Finally, we use the Heron formula to calculate the triangle area
[tex]A=\sqrt{s(s-a)(s-b)(s-c)}[/tex]
Where s is:
[tex]s=\frac{a+b+c}{2}[/tex]
Therefore
[tex]s=\frac{7.071+10+13.660}{2}[/tex]
[tex]s=15.37[/tex]
[tex]A=\sqrt{15.37(15.37-7.071)(15.37-10)(15.37-13.66)}[/tex]
[tex]A=34\ units^2[/tex]
