Question

find the length of side a.
a. 1
b. \sqrt{13}
c. \sqrt{61}
d. \sqrt{73}

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines states that for a triangle with sides \(a\), \(b\), \(c\) and angle \(A\) opposite side \(a\), \(a^{2}=b^{2}+c^{2}-2bc\cos A\). Here, \(b = 3\sqrt{2}\), \(c = 5\), and \(A=45^{\circ}\), \(\cos45^{\circ}=\frac{\sqrt{2}}{2}\).

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Step2: Simplify the expression

Calculate each term: \((3\sqrt{2})^{2}=9\times2 = 18\), \(5^{2}=25\), and \(2\times(3\sqrt{2})\times5\times\frac{\sqrt{2}}{2}=3\times5\times2= 30\).

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Step3: Solve for \(a\)

Take the square root of both sides. Since \(a>0\) (length of a side of a triangle), \(a=\sqrt{13}\).

Answer:

B. \(\sqrt{13}\)