If \( (\mathrm{F}), 8 \sqrt{3} \) are the magnitude of two forces act at a particle and thr measure of the angle between them is \( 120^{\circ} \), their resultant bisects the included angle between the two forces, then \( \mathrm{F}=\cdots \ldots \ldots \). newton. \( \begin{array}{llll}\text { (a) } 8 & \text { (b) } 8 \sqrt{3} & \text { (c) } 8 \sqrt{2}\end{array} \)

To find the value of \( F \) given that the resultant of two forces \( F \) and \( 8\sqrt{3} \) bisects the angle between them, we can use the law of cosines and the properties of the resultant force. Let: - \( F_1 = F \) - \( F_2 = 8\sqrt{3} \) - The angle between them \( \theta = 120^\circ \) The angle bisector theorem states that if a resultant force bisects the angle between two forces, then the ratio of the magnitudes of the two forces is equal to the ratio of the lengths of the segments created by the angle bisector on the opposite side. Using the law of cosines, the magnitude of the resultant \( R \) of two forces can be calculated as follows: \[ R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos(\theta)} \] Substituting the values we have: \[ R = \sqrt{F^2 + (8\sqrt{3})^2 + 2 F (8\sqrt{3}) \cos(120^\circ)} \] Since \( \cos(120^\circ) = -\frac{1}{2} \), we can substitute this into the equation: \[ R = \sqrt{F^2 + 192 - 8\sqrt{3} F} \] Now, since the resultant bisects the angle, we can also express the resultant in terms of the forces and the angle: \[ R = \frac{2 F_1 F_2}{F_1 + F_2} \] Substituting \( F_1 = F \) and \( F_2 = 8\sqrt{3} \): \[ R = \frac{2 F (8\sqrt{3})}{F + 8\sqrt{3}} = \frac{16\sqrt{3} F}{F + 8\sqrt{3}} \] Now we have two expressions for \( R \): 1. \( R = \sqrt{F^2 + 192 - 8\sqrt{3} F} \) 2. \( R = \frac{16\sqrt{3} F}{F + 8\sqrt{3}} \) Setting these equal to each other: \[ \sqrt{F^2 + 192 - 8\sqrt{3} F} = \frac{16\sqrt{3} F}{F + 8\sqrt{3}} \] Squaring both sides: \[ F^2 + 192 - 8\sqrt{3} F = \frac{(16\sqrt{3} F)^2}{(F + 8\sqrt{3})^2} \] Calculating the right side: \[ \frac{(16\sqrt{3} F)^2}{(F + 8\sqrt{3})^2} = \frac{768 F^2}{(F + 8\sqrt{3})^2} \] Now we have: \[ F^2 + 192 - 8\sqrt{3} F = \frac{768 F^2}{(F + 8\sqrt{3})^2} \] This equation can be solved for \( F \). However, we can also use the angle bisector property directly: From the angle bisector theorem, we have: \[ \frac{F}{8\sqrt{3}} = \frac{R}{R} \] This implies: \[ F = 8\sqrt{3} \cdot \frac{R}{R} = 8\sqrt{3} \] Thus, we can conclude that: \[ F = 8 \text{ newtons} \] So the answer is: \[ \boxed{8} \]