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Learning Math with Robotics
Midpoint Between Two Points in 3-Dimensions
Overview
In this lesson, students will learn how to apply the midpoint formula to two points in 3-dimensional space.
Learning Objectives
Students will be able to:
  • Calculate the midpoint between two points in 3-dimensional space
Lesson Description
In previous lessons, we discussed the midpoint formula for two points in 2-dimensional space. This formula can be extended for use in 3-dimensional space. 
 
Given the two endpoints \((x_1,y_1,z_1)\) and \((x_2,y_2,z_2)\), you can calculate the midpoint \((x_m,y_m,z_m)\) using the following formula:
\((x_m,y_m,z_m)=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2})\)

You can also use the midpoint3D code block (found in the dropdown menu of the midpoint code block) to calculate the distance between two points.
 
   
 
          
To use this block, input the x, y, and z coordinates of the first and second point. The block will then calculate the coordinates of the midpoint between those two points and store them in the variables x, y, and z. This is shown in Example 3.

 
Vocabulary
  • CS: midpoint3D Calculates the distance between two points in three dimensions.

 
Key Points
  1. The midpoint in 3D is the point halfway between two points in three-dimensional space.
  2. It is found by averaging the distances in x, y, and z direction of the two endpoints.
  3. This formula helps calculate the distance from the endpoints to the midpoint between two points in 3D space.
Warm-Up: 3D Midpoint Formula
Recall how to find the midpoint in 2-dimensional space. Given two points (2, 4, 6) and (8, 10, 12) in 3-dimensional space, how might the process change for finding the midpoint?
Example 1: Midpoint Formula in 3-Dimensions
Given the two endpoints \(P_1=(x_1,y_1,z_1)\) and \(P_2=(x_2,y_2,z_2)\) in 3-dimensional space, we can calculate the midpoint similarly as in 2-dimensional space. The midpoint \((x_m,y_m,z_m)\) is half way between the points \(P_1\) and \(P_2\) and can be calculated using the following formula:
\((x_m,y_m,z_m)=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2})\)
Example 2: Move the Spider to the Fly
A spider caught a fly at the midpoint of its web that stretches from the point (0,0,0) to the point (8,9,14) in a box. The code in the Workspace calculates the coordinates of the fly's position and determines how far the spider needs to crawl to reach the fly. Run the program and watch as the spider crawls 9.2331 inches. This makes sense because the midpoint is calculated to be
\((x_m,y_m,z_m)=(\frac{0+8}{2},\frac{0+9}{2},\frac{0+14}{2})=(\frac{8}{2},\frac{9}{2},\frac{14}{2})=(4,4.5,7)\)
and the distance from the point (0,0,0) to the point (4,4.5,7) is 
\(distance=\sqrt{(4)^2+(4.5)^2+(7)^2}=\sqrt{16+20.25+49}=\sqrt{85.25}=9.2231\)
Thus, we know the fly's position is (4,4.5,7) and the distance between the spider and the fly is 9.2331
Example 3: Calculate Midpoint Using the midpoint3D() Block
In the grid, we have two endpoints \(P_1=(0,0,0)\) and \(P_2=(6,4,9)\) in 3-dimensional space. Run the program to use the midpoint3D block to calculate the midpoint between these points. We see that the two points are  in apart.
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