1. **State the problem:** We are given points $A(-12,-9,k+3)$ and $B(-15,-9,3k)$, and the midpoint of segment $AB$ lies in the $xy$-plane. We need to find the value of $k$. 2. **Recall the midpoint formula:** The midpoint $M$ of segment $AB$ with coordinates $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$ is given by: $$M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2}\right)$$ 3. **Apply the midpoint formula:** $$M = \left(\frac{-12 + (-15)}{2}, \frac{-9 + (-9)}{2}, \frac{(k+3) + 3k}{2}\right) = \left(\frac{-27}{2}, \frac{-18}{2}, \frac{4k + 3}{2}\right) = \left(-13.5, -9, \frac{4k + 3}{2}\right)$$ 4. **Use the condition that the midpoint lies in the $xy$-plane:** The $xy$-plane means $z=0$, so: $$\frac{4k + 3}{2} = 0$$ 5. **Solve for $k$:** Multiply both sides by 2: $$4k + 3 = 0$$ Subtract 3: $$4k = -3$$ Divide both sides by 4: $$k = \frac{\cancel{4}k}{\cancel{4}} = \frac{-3}{4}$$ **Final answer:** $$k = -\frac{3}{4}$$