We are given a triangle with sides $a = 8$ cm, $b = 6$ cm, and the angle between them $C = 60^{\circ}$. We need to find the length of the third side, $c$, using the cosine rule.

1. Problem Description

We are given a triangle with sides $a = 8$ cm, $b = 6$ cm, and the angle between them $C = 60^{\circ}$. We need to find the length of the third side, $c$, using the cosine rule.

2. Solution Steps

The cosine rule states that

$c^2 = a^2 + b^2 - 2ab \cos(C)$

Substituting the given values:

$c^2 = 8^2 + 6^2 - 2 \times 8 \times 6 \times \cos(60^{\circ})$

We know that $\cos(60^{\circ}) = \frac{1}{2}$.

$c^2 = 64 + 36 - 2 \times 8 \times 6 \times \frac{1}{2}$

$c^2 = 100 - 48$

$c^2 = 52$

$c = \sqrt{52}$

$c = \sqrt{4 \times 13}$

$c = 2\sqrt{13}$

$c \approx 7.21$ cm

3. Final Answer

The length of the third side is $c = 2\sqrt{13}$ cm, which is approximately 7.21 cm.