Problem 17 - Precalculus

Question

find the midpoint of the line segment joining points \(P\) and \(Q\) (a) \(P(3,2)\) and \(Q(9,8)\) (b) \(P(-4,0)\) and \(Q(5,-3)\) (c) \(P(3,-6)\) and \(Q(-1,-2)\)

Step-by-Step Solution

Answer

(a) (6, 5), (b) (1/2, -3/2), (c) (1, -4)

Step 1: Understanding the Formula

To find the midpoint of a line segment joining two points \( P(x_1, y_1) \) and \( Q(x_2, y_2) \), use the formula: \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \). This gives the average of the x-coordinates and the y-coordinates of the points.

Step 2: Calculating Midpoint for Points (a)

For points \( P(3, 2) \) and \( Q(9, 8) \), apply the midpoint formula: \[ M = \left( \frac{3 + 9}{2}, \frac{2 + 8}{2} \right) = \left( \frac{12}{2}, \frac{10}{2} \right) = (6, 5) \]. So, the midpoint is \( (6, 5) \).

Step 3: Calculating Midpoint for Points (b)

For points \( P(-4, 0) \) and \( Q(5, -3) \), use the formula: \[ M = \left( \frac{-4 + 5}{2}, \frac{0 + (-3)}{2} \right) = \left( \frac{1}{2}, \frac{-3}{2} \right) \]. Therefore, the midpoint is \( \left( \frac{1}{2}, \frac{-3}{2} \right) \).

Step 4: Calculating Midpoint for Points (c)

For points \( P(3, -6) \) and \( Q(-1, -2) \), apply the formula: \[ M = \left( \frac{3 + (-1)}{2}, \frac{-6 + (-2)}{2} \right) = \left( \frac{2}{2}, \frac{-8}{2} \right) = (1, -4) \]. Thus, the midpoint is \( (1, -4) \).

Key Concepts

Line Segments

A line segment is a part of a line that is bounded by two distinct endpoints. It includes all the points on the line between these endpoints. Imagine stretching a piece of string between two pins on a board; the string is your line segment with pins marking the endpoints.
Understanding line segments is crucial when exploring geometry because they are the building blocks for more complex geometric shapes.
Key characteristics of line segments include:

• Fixed Length: Unlike a line, which extends infinitely in both directions, a line segment has a defined, fixed length.


• End Points: A line segment is denoted by its endpoints, typically named using capital letters such as P and Q.


• Straight Path: It follows a straight path, meaning its shortest path lies between the two endpoints.

Using the midpoint formula, you can easily find the point that divides the line segment into two equal parts. This is particularly useful in coordinate geometry for determining the center point between two given points. The midpoint's coordinates give the average of the coordinates of the endpoints, offering insight into how the two points are positioned relative to each other.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a mathematical discipline that uses a coordinate system to define and analyze the position of points in space. It's the bridge between algebra and geometry, allowing numerical solutions to geometric problems. In coordinate geometry, we work primarily with points, lines, and curves defined on the Cartesian plane.
Key Concepts of Coordinate Geometry include:

• Points: Defined by coordinates (x, y) on the Cartesian plane.


• Distance Formula: Used to calculate the distance between two points. It's given by \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).


• Midpoint Formula: Ideally used to find the point that bisects the line segment connecting two points, expressed as \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \).

Using these formulas and concepts, you can solve various geometry problems effectively, like finding the midpoint of a line segment, the length of the line, or the slope of the line connecting two points. These calculations have practical applications in fields like physics, engineering, and computer graphics.

Cartesian Plane

The Cartesian plane is a fundamental concept in mathematics, used widely in coordinate geometry. Named after René Descartes, it comprises a two-dimensional surface that makes it possible to measure geometric problems numerically.
The Cartesian plane consists of:

• Axes: Two perpendicular number lines intersecting at zero point (origin) divide the plane into four quadrants. These lines are known as the x-axis (horizontal) and y-axis (vertical).


• Origin: The point where the x-axis and y-axis meet is known as the origin, denoted as (0,0).


• Quadrants: The plane is divided into four quadrants, each accommodating signs of coordinates (positive or negative).

In practical applications, points on the Cartesian plane are expressed as coordinate pairs (x, y) which tell us precisely where the point lies relative to the origin. Understanding how these coordinates operate within each quadrant is vital for effectively using the midpoint formula in solving geometry problems. Knowing the precise location of each point allows you to understand the geometric relationships, such as those between line segments, and easily calculate midpoints and lengths.