In triangle $ABC$, side $a$ is $7$, side $b$ is $10$, and angle $A$ is $30°$. Angle $B$ is $45°$. Using the Law of Sines, find the length of side $c$. If necessary, write your answer in simplest radical form.

• A. $\frac{10\surd 6}{2}$
• B. $12$
• C. $\frac{7\surd 2}{2}$
• D. $\frac{7\surd 6}{2}$

Answer: (D)

Verified step by step guidance

1. Identify the given elements in triangle ABC: side a = 7, side b = 10, angle A = 30°, and angle B = 45°.
2. Recall the Law of Sines formula: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\), which relates the sides and their opposite angles.
3. Calculate angle C using the fact that the sum of angles in a triangle is 180°: \(C = 180^\circ - A - B = 180^\circ - 30^\circ - 45^\circ\).
4. Use the Law of Sines to find side c by setting up the ratio \(\frac{c}{\sin C} = \frac{a}{\sin A}\) and solve for c: \(c = \frac{a \sin C}{\sin A}\).
5. Substitute the known values of a, angle A, and angle C into the equation and simplify, expressing the answer in simplest radical form if necessary.