Question

Fill in the blank with the correct response.

Answer 1

4

Explanation:

Answer 2

x=4

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)= (adj)/(hyp) \\cos(\alpha)= (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)= (adj)/(hyp) \\cos(\alpha)= (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) = (2)/(x) \\cos(\alpha) = (x)/(8) \\(2)/(x) =(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.