Question

What are the missing sides and angles of this triangle? What is the area?

Answer 1

This does not appear to be a right triangle.  However, we know 2 sides and the included angle, so can find the unknown side length.  Let x represent this length.  Then:

x^2 = (9 m)^2 + (12 m)^2 - 2(9m)(12 m)*cos 30 degrees, or

x^2 = 81 + 144 - 216(sqrt(3) / 2).  Please solve for x^2 and then solve the result for x, making sure to choose the positive value.  The result will be the length of the side opposite the 30 degree angle.

With 1 of 3 angles known, and 3 of 3 sides known, you can use the Law of Sines to find the other two angles.  As a reminder, the Law of Sines looks like this:

   a            b              c
-------- = --------- = ----------
sin A       sin B       sin C.

You can give the 30-deg angle any name you want; then a, the length of the side opposite the 30-deg angle, which you have just found.  And so on.