Question

question 4 of 10 suppose a triangle has two sides of length 2 and 3 and that the angle between these two sides is 60°. what is the length of the third side of the triangle? a. 2√3 b. √7 c. √3 d. 2

Explanation:

Step1: Recall the law of cosines

Let the two given - side lengths be \(a = 2\), \(b = 3\), and the included - angle \(C=60^{\circ}\), and the unknown side be \(c\). The law of cosines formula is \(c^{2}=a^{2}+b^{2}-2ab\cos C\).

Step2: Substitute the values

We know that \(a = 2\), \(b = 3\), and \(\cos C=\cos60^{\circ}=\frac{1}{2}\). Substitute these values into the formula:
\[

$$\begin{align*} c^{2}&=2^{2}+3^{2}-2\times2\times3\times\frac{1}{2}\\ &=4 + 9-6\\ &=7 \end{align*}$$

\]

Step3: Solve for \(c\)

Take the square - root of both sides of the equation \(c^{2}=7\). Since \(c\) represents the length of a side of a triangle, \(c=\sqrt{7}\).

Answer:

B. \(\sqrt{7}\)