Let's begin with recalling the Law of Sines. If â–³ ABC has lengths of a, b, and c and angle measures of A, B, and C, then we can write that the ratios of the sine of an angle to the side opposite this angle are equal.
In our exercise we are given a triangle and asked to evaluate its perimeter. Let's take a look at the given picture and name vertices with consecutive letters.
The third angle has a measure of 63^(∘). Let's add this information to our diagram. Let a and b represent the lengths of AB and AC.
Since we are asked to evaluate the perimeter, we need to know all the side lengths. To do this, we will create a proportion using the Law of Sines.
sin 31^(∘)/a=sin 63^(∘)/9=sin 86^(∘)/b
Let's start with evaluating the value of a using cross multiplication.
Now we can evaluate the perimeter of the triangle by adding all side lengths. Remember that this will be an approximation as we are using approximate side lengths.
5.2+ 10.1+ 9=24.3
The perimeter is approximately 24.3 units.
