1. Find the midpoint of a line segment with endpoints at A(1,2,3) and B(4,5,6).

\((x_m,y_m,z_m) = (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2})\)
\((\frac{1 + 4}{2}, \frac{2 + 5}{2},\frac{3+6}{2}) = (\frac{5}{2},\frac{7}{2}, \frac{9}{2})\)

The midpoint of the line segment is \((\frac{5}{2}, \frac{7}{2}, \frac{9}{2})\).

2. The midpoint of a tunnel is M(5,8,11). If one endpoint is A(2,4,6), find the coordinates of the other endpoint B.

3. Find the midpoint of the line segment on the line given by z = 3x u2212 4 between the points (2, y1, z2) and (6, y1, z2) in 3D space. Assume that the line segment also extends vertically with y-coordinates such that y1 = 3 and y2 = 8.

4. If the midpoint of a line segment is (7,13,21) and one endpoint is at (19,15,21), where is the other endpoint?

5. A rectangular prism has vertices at (0,0,0) and (6,8,10). Find the coordinates of the center of the prism.

The center of the rectangular prism can be found by calculating the midpoint of the diagonal connecting (0,0,0) and (6,8,10).

\((x_m, y_m, z_m) = (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2})\)
\((\frac{0 + 6}{2}, \frac{0 + 8}{2}, \frac{0 + 10}{2}) = (\frac{6}{2}, \frac{8}{2}, \frac{10}{2})\)

The center of the prism is at (3,4,5).

6. Two support pillars of a bridge are located at points P1(5,10,0) and P2(15, 10, 20) in miles. To increase the bridge’s structural integrity, an additional pillar needs to be placed midway between the two existing ones. Where should this new pillar be placed?

7. Two seismic sensors are located at S1(10,20,5) and S2(30,60,15). For better earthquake detection, a new sensor needs to be placed exactly midway between these two. Determine where this new sensor should be placed.

8. A tight-rope walker ties a rope diagonally across a room to practice on. If the endpoints of the rope are at (16,10,10) feet and (2,8,0) feet, what is its midpoint?