Find the area of the triangle using the Law of Sines?

Question

Here is the problem: http://i6.tinypic.com/7y7kmx5.gif

a. 178
b. 7924
c. 9436
d. 18,872

Answer 1

The Law of Sines states:
a/SinA = b/sinB = c/sinC
where the lower case letter is the side opposite the angle with the capital letter.

Applying the Sine Rule:
m/sinM = n/sinN
212/sin93 = n/sin57 Note n is the side between MP
So n = sin57*212/sin93
n = 178.04

Now angle NMT = 33 (angles in triangle NTM = 180)
So angle PMT = 60 (93 - 33)

Using basic trig now:
cos60 = h / n (n is side MP)
So h = 178.04 * cos60
h = 89.02

So the area is half base times height
A = .5 * 89.02 * 212
A = 9436.23

So option C.

Answer 2

formula: total interrior angle is 180
angle C=180-93-57=30

A=b^2 sinA sinC/2sinB
=212^2 sin30 sin57 / 2 sin 93
=9436

Answer 3

By the law of sines:

MP/sin57° = NP/sin93°
MP = NP(sin57°/sin93°) = 212(sin57°/sin93°)