To solve this problem, we need to understand how gravity changes as we move inside the Earth.$\\$The weight of a body is given by $W = m \cdot g,$ where $g$ is the acceleration due to gravity.$\\$On the surface of the Earth, the weight of the body is $200$ N.$\\$The acceleration due to gravity at a distance $r$ from the center of the Earth is given by:$\\ g_r = g_0 \cdot \frac{r}{R}, \\$where $g_0$ is the acceleration due to gravity on the surface, $R$ is the radius of the Earth,$\\$and $r$ is the distance from the center of the Earth.$\\$Halfway down to the center of the Earth, $r = \frac{R}{2}. \\$Substitute into the equation for $g_r: \\ g_r = g_0 \cdot \frac{\frac{R}{2}}{R} = \frac{g_0}{2}. \\$The weight of the body halfway down is then:$\\ W_{half} = m \cdot g_r = m \cdot \frac{g_0}{2}. \\$Since the weight on the surface is $200$ N, we have:$\\ W_{surface} = m \cdot g_0 = 200$ N.$\\$Thus, $W_{half} = \frac{200}{2} = 100$ N.$\\$Therefore, the weight of the body halfway down to the center of the Earth is $100$ N.$\\$This corresponds to Option 4.