Question

Fill in the blank with the correct response.

Answer 1

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 $\alpha$, then we have for the small triangle on the left, the following trigonometric relationship:

$cos(\alpha)= \frac{adj}{hyp} \\cos(\alpha)= \frac{2}{x}$

because in that small triangle, the side adjacent (adj) to angle $\alpha$  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 $\alpha$ side "x":

$cos(\alpha)= \frac{adj}{hyp} \\cos(\alpha)= \frac{x}{8}$

Now, we can make both $cos(\alpha)$ expressions equal since they involve the same angle, and solve for "x" in the resultant formula:

$cos(\alpha) = \frac{2}{x} \\cos(\alpha) = \frac{x}{8} \\\frac{2}{x} =\frac{x}{8}\\16 = x^2$

therefore x is either "4" or "-4" to give 16 when squared.

We opt for the positive answer since we want a length.