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Suppose a triangle has two sides of length 2 and 3, and the angle between these sides is [tex]\(60^{\circ}\)[/tex]. What is the length of the third side?

A. [tex]\(\sqrt{7}\)[/tex]

B. [tex]\(\sqrt{3}\)[/tex]

C. [tex]\(2 \sqrt{3}\)[/tex]

D. 2


Sagot :

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]
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