Question

find the length of side a.

a. 7
b. \\( \sqrt { 47 } \\)
c. \\( \sqrt { 109 } \\)
d. 9

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines formula is \(a^{2}=b^{2}+c^{2}-2bc\cos A\). Here, \(b = 2\), \(c = 5\sqrt{3}\), and \(A=30^{\circ}\).
Substitute the values into the formula:
\(a^{2}=2^{2}+(5\sqrt{3})^{2}-2\times2\times5\sqrt{3}\times\cos30^{\circ}\)

Step2: Calculate each term

• Calculate \(2^{2}=4\).
• Calculate \((5\sqrt{3})^{2}=25\times3 = 75\).
• Calculate \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\), then \(2\times2\times5\sqrt{3}\times\frac{\sqrt{3}}{2}=30\).

So, \(a^{2}=4 + 75-30\).

Step3: Simplify the expression

\(a^{2}=49\).
Take the square root of both sides: \(a=\sqrt{49}=7\).

Answer:

A. 7