Question

find the length of side a.

a triangle abc with angle at a being 45 degrees, side ac = 5, side ab = 3√2. multiple choice options: a. √61, b. 1, c. √13, d. √13 (note: possible typo in original, but ocr as is)

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines formula is \( a^{2}=b^{2}+c^{2}-2bc\cos A\). Here, \(b = 3\sqrt{2}\), \(c = 5\), and \(A=45^{\circ}\), \(\cos45^{\circ}=\frac{\sqrt{2}}{2}\).
Substitute the values into the formula:
\(a^{2}=(3\sqrt{2})^{2}+5^{2}-2\times3\sqrt{2}\times5\times\frac{\sqrt{2}}{2}\)

Step2: Simplify each term

• Calculate \((3\sqrt{2})^{2}\): \((3\sqrt{2})^{2}=3^{2}\times(\sqrt{2})^{2}=9\times2 = 18\)
• Calculate \(5^{2}=25\)
• Calculate \(2\times3\sqrt{2}\times5\times\frac{\sqrt{2}}{2}\):

$$
LATEXBLOCK0
$$

Then \(a^{2}=18 + 25-30\)

Step3: Calculate \(a^{2}\) and \(a\)

\(a^{2}=18+25 - 30=13\), so \(a=\sqrt{13}\)

Answer:

D. \(\sqrt{13}\)