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Chapter 9: Problem 28

Find the coordinates of the midpoint of the line segment joining the points. (4,0,-6),(8,8,20)

Short Answer

Expert verified
The midpoint of the line segment joining the points (4,0,-6) and (8,8,20) is (6,4,7).

Step by step solution

01

Identify the coordinates of the two points

The two points are (4,0,-6) and (8,8,20).
02

Calculate the x-coordinate of the midpoint

To get the x-coordinate of the midpoint, add the x-coordinates of the two points and divide by 2. So, \( (4+8)/2 = 6\).
03

Calculate the y-coordinate of the midpoint

To get the y-coordinate of the midpoint, add the y-coordinates of the two points and divide by 2. So, \( (0+8)/2 = 4\).
04

Calculate the z-coordinate of the midpoint

To get the z-coordinate of the midpoint, add the z-coordinates of the two points and divide by 2. So, \( (-6+20)/2 = 7\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coordinate Geometry
Coordinate geometry is a branch of mathematics that uses numbers to represent geometric shapes. This method allows us to identify specific points on a grid, which is known as the Cartesian coordinate system.

The system is defined by an x-axis (horizontal), y-axis (vertical), and in three-dimensional space, a z-axis (perpendicular to both the x and y axes).

By using pairs or triplets of numbers called coordinates, we can precisely define the location of points. For example, the point (4,0,-6) tells us how far to move along each axis to reach that specific location. This makes it easier to analyze and solve geometric problems using algebraic methods.
Three-Dimensional Space
Understanding three-dimensional space is essential when working with coordinates that include three different values, which are represented as (x, y, z).

In this space, every point is identified with three coordinates, indicating its position concerning the three axes.

This third dimension, often referred to as the z-axis, adds depth to calculations and visualization. It expands basic two-dimensional geometry and provides a comprehensive way to model more complex structures, such as finding the midpoint of a line segment in 3D space.
Line Segments
A line segment is a part of a line contained between two endpoints, unlike a straight line which extends endlessly.

Finding the midpoint of a line segment is a common task in coordinate geometry. The midpoint is the exact center of the segment, located exactly halfway between the endpoints.
  • To find the midpoint, simply average the coordinates of the endpoints.
  • This involves adding the respective coordinates of the endpoints and then dividing by two.
  • The formulas: \(x_{mid} = \frac{x_1 + x_2}{2}\), \(y_{mid} = \frac{y_1 + y_2}{2}\), and \(z_{mid} = \frac{z_1 + z_2}{2}\) allow you to calculate each coordinate of the midpoint separately.


These calculated values enable you to identify the midpoint even in the context of three-dimensional space.

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Most popular questions from this chapter

Find \((\mathbf{a}) \mathbf{u} \cdot \mathbf{v},(\mathbf{b}) \mathbf{u} \cdot \mathbf{u},(\mathbf{c})\|\mathbf{u}\|^{2},(\mathbf{d})(\mathbf{u} \cdot \mathbf{v}) \mathbf{v}\) and \((e) u \cdot(2 v)\). $$ \mathbf{u}=\langle-4,8\rangle, \quad \mathbf{v}=\langle 6,3\rangle $$

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