In triangle $GHJ$, if $g$ (opposite angle $G$) is $5$ units, $h$ (opposite angle $H$) is $10$ units, and angle $G$ is $30$ degrees while angle $H$ is $60$ degrees, what is the length of line segment $hj$?

• A. $10$ units
• B. $15$ units
• C. $8.66$ units
• D. $5$ units

Answer: (C)

Verified step by step guidance

1. Identify the given elements in triangle GHJ: side g = 5 units opposite angle G = 30°, side h = 10 units opposite angle H = 60°, and we need to find side j opposite angle J.
2. Recall that the sum of angles in any triangle is 180°, so calculate angle J as: \(J = 180^\circ - G - H = 180^\circ - 30^\circ - 60^\circ\).
3. Use the Law of Sines, which states that \(\frac{g}{\sin G} = \frac{h}{\sin H} = \frac{j}{\sin J}\), to set up the proportion involving side j: \(\frac{j}{\sin J} = \frac{h}{\sin H}\).
4. Rearrange the equation to solve for j: \(j = h \times \frac{\sin J}{\sin H}\).
5. Substitute the known values of h, angle J, and angle H into the equation and compute the sine values to find the length of side j.