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

A. [tex]$\sqrt{3}$[/tex]
B. [tex][tex]$2 \sqrt{3}$[/tex][/tex]
C. [tex]$\sqrt{7}$[/tex]
D. 2



Answer :

To find the length of the third side of a triangle when we know the lengths of two sides and the angle between them, we can use the Law of Cosines. The Law of Cosines formula is given by:

[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(\theta) \][/tex]

where:
- [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the lengths of the two known sides,
- [tex]\( \theta \)[/tex] is the included angle,
- [tex]\( c \)[/tex] is the length of the third side.

In this problem, we know:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = 3 \)[/tex]
- [tex]\( \theta = 60^\circ \)[/tex]

First, we need to convert the angle from degrees to radians, since the cosine function in the Law of Cosines formula often requires the angle in radians.

[tex]\[ \theta = 60^\circ = \frac{\pi}{3} \][/tex]

Next, we apply the Law of Cosines formula:

[tex]\[ c^2 = 2^2 + 3^2 - 2 \cdot 2 \cdot 3 \cdot \cos\left(\frac{\pi}{3}\right) \][/tex]

We know that:

[tex]\[ \cos\left(\frac{\pi}{3}\right) = 0.5 \][/tex]

Therefore, substituting these values into the formula gives:

[tex]\[ c^2 = 4 + 9 - 2 \cdot 2 \cdot 3 \cdot 0.5 \][/tex]

[tex]\[ c^2 = 4 + 9 - 6 \][/tex]

[tex]\[ c^2 = 7 \][/tex]

To find [tex]\( c \)[/tex], we take the square root of both sides:

[tex]\[ c = \sqrt{7} \][/tex]

Thus, the length of the third side of the triangle is:

[tex]\[ \boxed{\sqrt{7}} \][/tex]