Answer:
x=4
Step-by-step explanation:
Notice that the angle between side x and side 2 in the small right triangle on the left is the same angle between the sides x and (2+6 = 8) in the larger right angle triangle.
if we name the angle in question [tex]\alpha[/tex], then we have for the small triangle on the left, the following trigonometric relationship:
[tex]cos(\alpha)= \frac{adj}{hyp} \\cos(\alpha)= \frac{2}{x}[/tex]
because in that small triangle, the side adjacent (adj) to angle [tex]\alpha[/tex] is "2" and the hypotenuse (hyp) is "x" .
We can set some similar relationship for the larger triangle, which has a hypotenuse (hyp) of length (2+6 = 8), and that has for side adjacent (adj) to the angle [tex]\alpha[/tex] side "x":
[tex]cos(\alpha)= \frac{adj}{hyp} \\cos(\alpha)= \frac{x}{8}[/tex]
Now, we can make both [tex]cos(\alpha)[/tex] expressions equal since they involve the same angle, and solve for "x" in the resultant formula:
[tex]cos(\alpha) = \frac{2}{x} \\cos(\alpha) = \frac{x}{8} \\\frac{2}{x} =\frac{x}{8}\\16 = x^2[/tex]
therefore x is either "4" or "-4" to give 16 when squared.
We opt for the positive answer since we want a length.
