Final answer:
Using the Law of Sines and the given angles and side length, we find that the length of side r in triangle ARST is approximately 2.5 inches to the nearest tenth.
Step-by-step explanation:
To find the length of side r in triangle ARST, given that side t = 9 - 5 inches, and the angles ∠ZR = 149° and ∠S = 12°, we can use the Law of Sines. The Law of Sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle. In this case, the triangle ARST is not fully described, but it seems that ∠T might be the remaining angle, which can be calculated as:
∠T = 180° - ∠ZR - ∠S
∠T = 180° - 149° - 12° = 19°
Now we use the Law of Sines:
sin(∠T) / t = sin(∠S) / r
To find r, we rearrange the formula:
r = (t * sin(∠S)) / sin(∠T)
Since we know t = 4 inches (9 - 5), we plug in the values:
r = (4 * sin(12°)) / sin(19°)
Finally, calculate r using a calculator and rounding to the nearest tenth:
r ≈ (4 * 0.2079) / 0.3256 ≈ 2.5 inches
Thus, to the nearest tenth of an inch, the length of side r is approximately 2.5 inches.