We are asked to find the value of x in the given figure. Let's take a look at the given picture. We will label the vertices with consecutive letters.
As we can see, △ ABC is isosceles, so BC is also 6. Since we have all side lengths of △ ABC we can find the measure of ∠C by using the Law of Cosines.
AB^2=BC^2+AC^2-2(BC)(AC)cos CBy substituting the appropriate side lengths, we will solve the above equation.
Now we can evaluate the measure of ∠C using the inverse cosine.
cos C=34.81/70.8 ⇓
C=cos ^(-1)34.81/70.8≈ 60.55^(∘)
The measure of ∠C is approximately 60.55^(∘). Let's add this information to our picture.
Now, we will use the Law of Sines to write a proportion for ∠BCD. According to this law, in a triangle the ratio of the sine of an angle to the opposite side length is constant.
sin 68^(∘)/6=sin 60.55^(∘)/x
Let's solve this proportion using cross multiplication.
