Let's begin with recalling the Law of Cosines. If â–³ ABC has lengths of a, b, and c and angle measures of A, B, and C, then we can write equations that relate the side lengths of this triangle and the cosine of one of its angles.
Now let's take a look at the given picture. We are asked to evaluate the perimeter of the given triangle, so we need to know all the side lengths. Let x represent the length of the missing side.
Using the Law of Cosines, we can write an equation for x.
x^2= 88^2+ 152^2-2( 88)( 152)cos 39^(∘)
Let's solve the equation. Notice that since x represents a length, we will consider only the positive case when taking a square root of x^2.
The length of the missing side is approximately 100.3. Now we can find the perimeter of the triangle by adding all side lengths. Notice that this will be approximation as we are using an approximate side length.
P= 100.3+ 88+ 152=340.3
The perimeter of this triangle is approximately 340.3.
