Mid point of two points in 3D Space
Learning objective
1. Understand the concept of finding the midpoint between two points in three-
dimensional (3D) space.
2. Apply the midpoint formula to determine the coordinates of the midpoint
between two points.
3. Solve problems involving midpoints in 3D coordinates.
Lesson Content
1. Introduction to Midpoints in 3D Space
In 3D space, the midpoint of a line segment joining two points is the point that lies exactly halfway between them. Just like in two-dimensional space, the midpoint is useful in various applications, such as geometry, modeling, and vector analysis.
2. Definition: Midpoint in 3D Space
The midpoint of a line segment joining two points \( A(x_1, y_1, z_1) \) and \( B(x_2, y_2, z_2) \) in 3D space is the point \( M(x_m, y_m, z_m) \) that divides the line segment into two equal parts.
3. Formula for Finding the Midpoint
The formula to find the coordinates of the midpoint \( M(x_m, y_m, z_m) \) between two points \( A(x_1, y_1, z_1) \) and \( B(x_2, y_2, z_2) \) is:
\[
M(x_m, y_m, z_m) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right)
\]
This formula calculates the average of the x-coordinates, y-coordinates, and z-coordinates of the two points to determine the midpoint.
4. Worked Examples
Example 1:
Find the midpoint between the points \( A(2, 3, 5) \) and \( B(6, 7, 9) \).
Solution:
Using the midpoint formula:
\[
M(x_m, y_m, z_m) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right)
\]
Substitute the coordinates of \( A \) and \( B \):
\[
M = \left( \frac{2 + 6}{2}, \frac{3 + 7}{2}, \frac{5 + 9}{2} \right)
\]
\[
M = \left( \frac{8}{2}, \frac{10}{2}, \frac{14}{2} \right)
\]
\[
M = (4, 5, 7)
\]
So, the midpoint between \( A(2, 3, 5) \) and \( B(6, 7, 9) \) is \( (4, 5, 7) \).
Example 2:
Find the midpoint between the points \( C(-1, -3, 4) \) and \( D(3, -1, -2) \).
Solution:
Using the midpoint formula:
\[
M(x_m, y_m, z_m) = \left( \frac{-1 + 3}{2}, \frac{-3 + (-1)}{2}, \frac{4 + (-2)}{2} \right)
\]
\[
M = \left( \frac{2}{2}, \frac{-4}{2}, \frac{2}{2} \right)
\]
\[
M = (1, -2, 1)
\]
So, the midpoint between \( C(-1, -3, 4) \) and \( D(3, -1, -2) \) is \( (1, -2, 1) \).
5. Practice Problems
1. Find the midpoint between the points \( P(1, 4, 7) \) and \( Q(5, 8, 12) \).
2. Calculate the midpoint between the points \( R(0, 0, 0) \) and \( S(3, 6, 9) \).
3. Find the midpoint between \( T(-2, -3, 1) \) and \( U(2, 0, -1) \).
4. If the midpoint of the points \( A(4, 2, 6) \) and \( B(x, 7, 9) \) is \( (5, 5, 7) \), find the value of \(x\).
Answers to Practice Problems
Question no:1
Find the midpoint between the points \( P(1, 4, 7) \) and \( Q(5, 8, 12) \).
Question no:2
Calculate the midpoint between the points \( R(0, 0, 0) \) and \( S(3, 6, 9) \).
Question no:3
Find the midpoint between \( T(-2, -3, 1) \) and \( U(2, 0, -1) \).
Question no:4
If the midpoint of the points \( A(4, 2, 6) \) and \( B(x, 7, 9) \) is \( (5, 5, 7) \), find the value of \(x\).
6. Summary
The midpoint of a line segment in 3D space is found by averaging the coordinates of the two points.
– The formula for the midpoint between two points \( A(x_1, y_1, z_1) \) and \( B(x_2, y_2, z_2) \) is:
\[
M(x_m, y_m, z_m) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right)
\]