Before we can set up a proportion, we need to know the third side of the larger triangle, since that side is actually the one that corresponds to 6, since both are the shortest sides in their respecitve triangles. We can solve for that side using the Pythagorean Theorem,

subtract 100 from both sides

take the square root

Now we can set up a proportion comparing corresponding sides to solve for x:

 cross-multiply

divide by 7.5

The Law of Cosines is as follows:

Notice these equations contains the Pythagorean Theorem, , within it.

The term at the end is the adjusting term for triangles which are not right triangles. 

This is a prime example of a case that calls for using the Law of Cosines, which states

where , , and  are the three sides of the triangle, and  is the angle opposite side .  Looking at our triangle, taking , then we have , , and .  Plugging this into our formula, we get.

Using our calculator to approximate the cosine value gives

Simplifying further gives 

Solving by taking the square root gives

To apply the Law of Cosines,  is the unknown,  and  are the respective given sides, and the given angle is .

Therefore, the equation becomes: 

Which yields 

Add  to the other two given sides to get the perimeter, 

We can solve for x using the law of cosines, where C is the angle between sides a and b.

In this case:

The Law of Cosines come in different forms depending on which angle or side you wish to find. One of the missing bits of information about our triangle is side length a. It is important to find this side because with side length a we can use the Law of Sines to easily find the angle measures. Side a "unlocks" the problem.

The pertinent LOC is .

Now that we know side a, we can use the reciprocal form of the Law of Sines to find the remaining angle measures.

Angle B:

To find the corresponding angle we take the inverse sine. 

But there are two angles between 0° and 180°; there is 44.7° and . How do we know which angle to choose? We find out by solving for the last angle C with both of our hypothetical angles for angle B. Since side c is the largest side, it follows it should have the largest angle of all three angles in the triangle. Compute the measure of angle C by subtracting the given angle (angle A) and the angle we calculated (angle B) from 180°. Do this once with 44.7° and once with 135.3°. The first case results in the largest angle C and fits with c being the largest side. Thus angle B=44.7° and angle C must equal 110.3°.

To find an angle in an oblique triangle where all sides are known, use the law of cosines:

The law of cosines states that .

So:

The sketch of the situation below shows that the angle between the ships from the radar station is 65 degrees.

To find the distance between the ships, use the law of cosines:

To find , you must first find side c using the law of cosines:

Knowing c, you can find  using the law of sines or the law of cosines.

Law of sines:

Law of cosines: