Sure, let's solve the question step-by-step. Given:
- Two sides of the triangle are [tex]\( a = 2 \)[/tex] and [tex]\( b = 3 \)[/tex].
- The angle between these two sides is [tex]\( \theta = 60^\circ \)[/tex].
To find the third side [tex]\( c \)[/tex], we'll use the cosine rule which states:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(\theta) \][/tex]
Let's break this down:
1. Calculate [tex]\( a^2 \)[/tex] and [tex]\( b^2 \)[/tex]:
- [tex]\( a^2 = 2^2 = 4 \)[/tex]
- [tex]\( b^2 = 3^2 = 9 \)[/tex]
2. Calculate [tex]\( 2ab \cos(\theta) \)[/tex]:
- [tex]\( 2ab = 2 \times 2 \times 3 = 12 \)[/tex]
- [tex]\( \cos(60^\circ) = \frac{1}{2} \)[/tex]
- Therefore, [tex]\( 2ab \cos(60^\circ) = 12 \times \frac{1}{2} = 6 \)[/tex]
3. Substitute these values into the cosine rule equation:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(\theta) \][/tex]
[tex]\[ c^2 = 4 + 9 - 6 \][/tex]
4. Simplify the equation:
[tex]\[ c^2 = 4 + 9 - 6 = 7 \][/tex]
5. Solve for [tex]\( c \)[/tex]:
[tex]\[ c = \sqrt{7} \][/tex]
Hence, the length of the third side of the triangle is [tex]\( \sqrt{7} \)[/tex]. The correct answer is:
A. [tex]\( \sqrt{7} \)[/tex]
