7). RE=?

In triangle ROS, angle O is 90 degrees and angle S is 60 degrees. Therefore, angle R is 30 degrees. We are given SO = 7. In triangle ROS, tan(60) = SO/RO, so RO = SO/tan(60) = 7 / sqrt(3). In triangle RES, angle RES = 180 - 60 = 120 degrees. Angle R = 30 degrees. Angle RSE = 180 - 120 - 30 = 30 degrees. Thus, triangle RES is isosceles with RE = SE. In triangle ROS, sin(60) = RO/RS, so RS = SO/sin(60) = 7 / (sqrt(3)/2) = 14/sqrt(3). In triangle RES, by the Law of Sines, RE/sin(30) = RS/sin(120). RE = RS * sin(30) / sin(120) = (14/sqrt(3)) * (1/2) / (sqrt(3)/2) = (14/sqrt(3)) * (1/sqrt(3)) = 14/3.