Answer:
Option (C) 6.736
Step-by-step explanation:
Please look at the attached modified schematic, where I have constructed a perpendicular line to the base of the triangle.
This creates two Right triangles (i.e. one angle will always be 90°). In doing so, the 80° angle is divided into 30° and 50° angles. In doing so we also then know that on the left Right-triangle the bottom left angle will be 60° since for any Triangle all three angles must add up to 180°.
(Check your self: On the left triangle we have 90° + 30° + 60° = 180° and on the right triangle we have 90° + 50° + 40° = 180°, so we are correct).
Now let us first find the value of side named [tex]y[/tex] on the figure using Right-triangle Trigonometry. In a Right-triangle the longest side is called the hypotenuse (here denoted by the sides [tex]x[/tex] and the one valued [tex]5[/tex]).
So we have:
[tex]sin(60)=\frac{opposite side}{hypotenuse}= \frac{y}{5}\\[/tex]
so solving for [tex]y[/tex] gives:
[tex]sin(60)=\frac{y}{5}\\ y=5sin(60)\\y=\frac{5\sqrt{3} }{2}[/tex]
Similarly for the other triangle we can write:
[tex]sin(40)=\frac{y}{x}\\\\[/tex]
[tex]sin(40)=\frac{\frac{5\sqrt{3} }{2} }{x}\\[/tex]
[tex]x=\frac{\frac{5\sqrt{3} }{2} }{sin(40)}\\\\x=6.736[/tex]
So the correct option is Option (C) 6.736
