To find the length of the third side of a triangle when two sides and the included angle are given, we can use the Law of Cosines. The Law of Cosines states:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(C) \][/tex]
In this problem, we are given:
- Side [tex]\( a = 2 \)[/tex]
- Side [tex]\( b = 3 \)[/tex]
- Angle [tex]\( C = 60^\circ \)[/tex]
Let's plug these values into the Law of Cosines formula.
1. First, we calculate the cosine of the given angle:
[tex]\[ \cos(60^\circ) = \frac{1}{2} \][/tex]
2. Next, substitute this value along with the lengths of sides [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the formula:
[tex]\[ c^2 = 2^2 + 3^2 - 2 \cdot 2 \cdot 3 \cdot \cos(60^\circ) \][/tex]
3. This simplifies to:
[tex]\[ c^2 = 4 + 9 - 2 \cdot 2 \cdot 3 \cdot \frac{1}{2} \][/tex]
4. Further simplifying:
[tex]\[ c^2 = 13 - 6 \][/tex]
5. So:
[tex]\[ c^2 = 7 \][/tex]
6. To find the length of side [tex]\( c \)[/tex], take the square root of both sides:
[tex]\[ c = \sqrt{7} \][/tex]
Therefore, the length of the third side of the triangle is [tex]\( \sqrt{7} \)[/tex].
So the correct answer is:
A. [tex]\( \sqrt{7} \)[/tex]
