Let's begin with recalling the Law of Sines. If △ 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 quadrilateral and asked to evaluate its perimeter. Let's take a look at the given picture. We will label the missing diagonal, sides, and angles.
We will start with evaluating the value of x.
x+ 98^(∘)+ 27^(∘)=180^(∘) ⇓
x=55^(∘)
Next, we will find the value of y.
y+40^(∘)+68^(∘)=180^(∘) ⇓
y=72^(∘)
Let's add this information to our picture.
Next, we can find the values of a and d.
To find these values, we will create a proportion using the Law of Sines.
sin 55^(∘)/a=sin 27^(∘)/2.3=sin 98^(∘)/d
Let's start with evaluating the value of a using cross multiplication.
The next step will be to evaluate the values of b and c.
To do this, we will create a proportion using the Law of Sines.
sin 40^(∘)/b=sin72^(∘)/5=sin68^(∘)/c
First we will evaluate the value of b using cross multiplication.
Now we can evaluate the perimeter of the quadrilateral by adding all the side lengths.
Remember that this will be an approximation as we are using approximate side lengths.
4.1+ 2.3+3.4+4.9=14.7
The perimeter of this figure is approximately 14.7 units.
