Problem 36 - Precalculus with Calculus Previews

Question

Find the coordinates of the midpoint of the line segment between the given points. $$ (0,5,-8),(4,1,-6) $$

Step-by-Step Solution

Answer

The midpoint is at (2, 3, -7).

Step 1: Identify the coordinates

We have two points given: Point A with coordinates \((0, 5, -8)\) and Point B with coordinates \((4, 1, -6)\).

Step 2: Use the midpoint formula

The formula for finding the midpoint \(M\) of a line segment joining two points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) is:\[\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right)\]

Step 3: Plug the coordinates into the formula

Using the coordinates of the points: \((x_1, y_1, z_1) = (0, 5, -8)\) and \((x_2, y_2, z_2) = (4, 1, -6)\), we plug them into the formula.\[M = \left( \frac{0 + 4}{2}, \frac{5 + 1}{2}, \frac{-8 + (-6)}{2} \right)\]

Step 4: Calculate the midpoint

Calculate the numerical result of the formula:- Calculate the x-coordinate: \( \frac{0 + 4}{2} = \frac{4}{2} = 2 \)- Calculate the y-coordinate: \( \frac{5 + 1}{2} = \frac{6}{2} = 3 \)- Calculate the z-coordinate: \( \frac{-8 + (-6)}{2} = \frac{-14}{2} = -7 \)Hence, the midpoint \(M\) is \((2, 3, -7)\).

Key Concepts

Coordinates

Coordinates are a way to describe positions and points in mathematical space. They provide a numeric representation of where a point is located. In a 2D space, coordinates have two values, typically written as \(x, y\), where:

• \(x\) is the horizontal position
• \(y\) is the vertical position

In 3D geometry, however, things are a bit more complex. We use an additional value to account for the third dimension, making it a trio named \(x, y, z\). Here:

• \(x\): the position along the horizontal (left to right)
• \(y\): the position along the vertical (front to back)
• \(z\): the position in the depth (up to down)

With coordinates in 3D space, we can precisely pinpoint any location, like plotting Point A and Point B given in the exercise.

3D geometry

Three-dimensional (3D) geometry involves the study of shapes and figures in a space that has three dimensions. In contrast to 2D geometry, which only considers width and height, 3D geometry also considers depth. This gives a more realistic representation of objects just like we encounter in the real world.

Here's how the concepts of 3D geometry relate to our exercise:

• We have two points in space: \((0, 5, -8)\) and \((4, 1, -6)\)
• These points are defined in terms of their coordinates, each having three values: \(x, y,\) and \(z\)
• The task is to find a particular point, the midpoint, which expresses a position exactly between these two points in 3D space

Understanding 3D geometry involves recognizing how objects extend into three dimensions, which is crucial when working with problems like finding midpoints and other spatial calculations.

Line Segment

A line segment is a part of a line that is bounded by two distinct end points. It contains every point on the line between its endpoints. In geometry, line segments are particularly useful to measure distances and define midpoints, like in this exercise.

Let's break down the line segment with the given points:

• We begin with two endpoints: Point A \((0, 5, -8)\) and Point B \((4, 1, -6)\)
• A line segment exists between these two points, spanning in 3D space
• The midpoint of this line segment is the point directly in the middle between Point A and Point B

Finding the midpoint is key in dividing a line segment equally into two smaller segments. This point is calculated using the midpoint formula provided in the step-by-step solution. The formula helps in deducing the exact coordinates that bisect the line segment, ensuring both segments remain equal in length.