8 14
А
34°
11
b
A
b = 578.36
B
b = 7.85
С
b = 61.66
D
b = 24.05

9514 1404 393

  B.  7.85

Step-by-step explanation:

Your knowledge of the triangle inequality tells you 'b' cannot be greater than 11+14 = 25, eliminating choices A and C. If 'b' were 24.05, as suggested by choice D, the angle at B would be nearly a straight angle.

So, the only viable choice is b = 7.85.

__

Your knowledge of "special" triangles tells you that if B were 30°, then b would be 14/2 = 7.0. The angle is slightly larger than that, so the side opposite is slightly larger than 7.0.

__

In this geometry, the Law of Cosines would be used to find b. That relationship tells you ...

  b = √(a^2 +c^2 -2ac·cos(B))

  b = √(121 +196 -308·cos(34°)) ≈ √61.66 ≈ 7.85

_____

About special triangles

There are two "special" right triangles whose trig functions are not difficult to remember. They appear quite often in algebra, geometry, and trig problems, so remembering their properties can be useful.

A 30°-60°-90° right triangle has side lengths in the ratios 1 : √3 : 2. The shortest side is exactly half the length of the hypotenuse. This fact is what we referred to above.

A 45°-45°-90° isosceles right triangle has side lengths in the ratios 1 : 1 : √2. This is the triangle you get by drawing a diagonal in a square. These ratios tell you the diagonal of a square is always √2 times the side length.