SOLUTION: Two angles of an oblique triangle are 60 and 50 degrees and their included side is 400 cm long. Find the area. Use sine law to determine the remaining sides.

Algebra ->  Parallelograms -> SOLUTION: Two angles of an oblique triangle are 60 and 50 degrees and their included side is 400 cm long. Find the area. Use sine law to determine the remaining sides.      Log On


   

Question 1150871: Two angles of an oblique triangle are 60 and 50 degrees and their included side is 400 cm long. Find the area. Use sine law to determine the remaining sides.
Found 2 solutions by josgarithmetic, jim_thompson5910:
Answer by josgarithmetic(39625) About Me  (Show Source):
You can put this solution on YOUR website!
As a start, the missing angle opposite the 400 cm side is 180-60-50=70 degrees.

Law of sines should give equations L%2Fsin%2850%29=400%2Fsin%2870%29 and R%2Fsin%2860%29=400%2Fsin%2870%29. That is taking L is opposite of the 50 degree angle,...

L is opposite the 50 degree angle and R is opposite the 60 degree angle.
system%28L=326.08%2CR=368.64%29

y, for height
y%2FL=sin%2860%29
y=326.08%2Asin%2860%29
y=282.3935approximately

AREA,
%281%2F2%29%2A400%2A282.3935
Round the result as needed.

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

The tutor @josgarithmetic has the correct answers. This is what the triangle looks like


An alternative to finding the height (to find the area of the triangle) is to use the formula below
A = 0.5*p*q*sin(R)
A = area of triangle
R is the angle between sides p & q.
This can be thought of as the SAS triangle area formula because we have two sides (p & q) and the included angle between them (angle R). All three of which help us find the area.

So lets say we have
p = 400
q = 368.64
R = 50
So,
A = 0.5*p*q*sin(R)
A = 0.5*400*368.64*sin(50)
A = 56478.924702276

The much more closer approximation of the area of the triangle is 56479.2302073497 which is fairly close to 56478.924702276. The discrepancy is due to the number of decimal digits we used for side q = 368.64

------------------------------------------------
Here is more accurate look at the triangle (with better approximations for the side lengths with decimal values)


Let's recompute the area with q = 368.6419940428
p = 400
q = 368.6419940428
R = 50
A = 0.5*p*q*sin(R)
A = 0.5*400*368.6419940428*sin(50)
A = 56479.2302073572
This is much closer to 56479.2302073497

------------------------------------------------

There is yet another way to find the area of a triangle if you know two angles and the included side. This can be informally known as the ASA triangle area rule. ASA stands for "angle side angle". Based on the information you are given, the ASA triangle area formula will be much more quicker and direct to finding the area.

There are 3 slightly different variations of the ASA rule. The rule we'll use is
Area+=+%28c%5E2%2Asin%28A%29%2Asin%28B%29%29%2F%282%2Asin%28C%29%29
where A,B,C are the angles and 'a','b','c' are the three sides opposite those respective angles.

Refer to the diagram above
A = 50
B = 60
C = 70
a = 326.0829876384
b = 368.6419940428
c = 400

So,
Area+=+%28c%5E2%2Asin%28A%29%2Asin%28B%29%29%2F%282%2Asin%28C%29%29

Area+=+%28400%5E2%2Asin%2850%29%2Asin%2860%29%29%2F%282%2Asin%2870%29%29

Area+=+56479.2302073497
We get a much more accurate value here (though it's still an approximation).