Question

Find the midpoint of the line segment joining the points \(P_{1}\) and \(P_{2}\). $$ P_{1}=(-4,-3) ; \quad P_{2}=(2,2) $$

Short answer

The midpoint is \((-1, -0.5)\).

Step-by-step solution

Identify the coordinates of the points

The points given are \( P_{1} = (-4, -3) \) and \( P_{2} = (2, 2) \). These coordinates represent the endpoints of the line segment.

Use the midpoint formula

The formula to find the midpoint \(M\) of a line segment joining two points \((x_{1}, y_{1})\) and \((x_{2}, y_{2})\) is: \[ M = \left( \frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2} \right) \]

Substitute the coordinates into the formula

Substituting \(x_{1} = -4\), \(y_{1} = -3\), \(x_{2} = 2\), and \(y_{2} = 2\) into the formula: \[ M = \left( \frac{-4 + 2}{2}, \frac{-3 + 2}{2} \right) \]

Simplify the expressions

Perform the arithmetic operations to simplify: \[ M = \left( \frac{-2}{2}, \frac{-1}{2} \right) = (-1, -0.5) \]

Key concepts

Midpoint of a Line Segment

The midpoint of a line segment is the point that is exactly halfway between the two endpoints of the segment. Imagine you have a line segment with endpoints at coordinates \(P_{1} = (-4, -3)\) and \(P_{2} = (2, 2)\). The midpoint, \(M\), is the average of the x-coordinates and the y-coordinates of the endpoints. To find \(M\), you use the midpoint formula:
\[ M = \left( \frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2} \right) \]
This formula tells us that you need to:

• Add the x-coordinates of \P_{1}\ and \P_{2}\, then divide by 2.
• Add the y-coordinates of \P_{1}\ and \P_{2}\, then divide by 2.

For our points, \(P_{1} = (-4, -3) \) and \(P_{2} = (2, 2) \), plug the coordinates into the formula:
\[ M = \left( \frac{-4 + 2}{2}, \frac{-3 + 2}{2} \right) = (-1, -0.5) \]
Thus, the midpoint of the segment joining \(P_{1}\) and \(P_{2}\) is at \((-1, -0.5)\).

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This method allows you to use algebraic techniques to solve geometric problems.
Points in a plane are described using ordered pairs of numbers, known as coordinates \((x, y)\). These coordinates are measured along the x-axis (horizontal) and y-axis (vertical). For example, the point \(P_{1} = (-4, -3)\) is 4 units left of the origin and 3 units down. Similarly, \(P_{2} = (2, 2)\) is 2 units right and 2 units up.
Using coordinate geometry, you can:

• Find the distance between two points.
• Determine the slope of a line.
• Calculate the midpoint of a line segment.

Coordinate geometry combines the precision of algebra with the visual insights of geometry to solve problems effectively.

Arithmetic Operations

Arithmetic operations are basic mathematical operations that include addition, subtraction, multiplication, and division. These are essential tools in concepts like finding the midpoint of a line segment.
For the midpoint formula:

• First, add the x-coordinates of the endpoints: \(-4 + 2 = -2\).
• Second, add the y-coordinates of the endpoints: \(-3 + 2 = -1\).
• Finally, divide each sum by 2: \(\frac{-2}{2} = -1\) and \(\frac{-1}{2} = -0.5 \).

Knowing how to perform these operations is crucial. They help simplify the calculations in the midpoint formula, leading to finding the midpoint coordinates accurately.
Here's a quick recap:

• Add the coordinates.
• Then divide by 2.

These steps are simple but important!