Distance and Midpoint Formulas in the Cartesian Plane

Graphs and the Cartesian Plane

Rectangular (Cartesian) Coordinate System

The Cartesian coordinate system is a two-dimensional plane defined by a horizontal axis (the x-axis) and a vertical axis (the y-axis). The intersection of these axes is called the origin, denoted as (0, 0). Points are located using ordered pairs (x, y), where x is the abscissa (horizontal value) and y is the ordinate (vertical value). The plane is divided into four quadrants based on the signs of x and y.

• Quadrant I: x > 0, y > 0

• Quadrant II: x < 0, y > 0

• Quadrant III: x < 0, y < 0

• Quadrant IV: x > 0, y < 0

Moving right from the origin increases the x-coordinate (positive abscissa), moving left decreases it (negative abscissa). Moving up increases the y-coordinate (positive ordinate), moving down decreases it (negative ordinate).

Plotting Points and Graphs

To plot points, use the ordered pair (x, y) to locate the position on the grid. The grid is divided by the axes and labeled with positive and negative values.

This grid is used to visualize points, line segments, and geometric relationships in the coordinate plane.

Distance Formula

Definition and Application

The distance formula is used to compute the length between two points in the coordinate plane. It is derived from the Pythagorean Theorem, where the distance is the hypotenuse of a right triangle formed by the horizontal and vertical differences between the points.

• Formula: The distance between points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

$\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Steps:

  1. Find the difference in x-coordinates: $\Delta x = x_2 - x_1$

  2. Find the difference in y-coordinates: $\Delta y = y_2 - y_1$

  3. Apply the formula above to calculate the distance.

• Example: Find the distance between $(-3, -5)$ and $(3, 3)$:

  • $\Delta x = 3 - (-3) = 6$

  • $\Delta y = 3 - (-5) = 8$

  • Distance: $\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$

Application: The distance formula is used in geometry, navigation, and contextual problems involving locations and measurements.

Midpoint Formula

Definition and Application

The midpoint formula determines the point exactly halfway between two given coordinates. The midpoint is equidistant from both endpoints and is found by averaging the x and y values separately.

• Formula: The midpoint between $(x_1, y_1)$ and $(x_2, y_2)$ is:

$\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

• Steps:

  1. Add the x-coordinates and divide by 2.

  2. Add the y-coordinates and divide by 2.

• Example: Find the midpoint between $(-6, -6)$ and $(-5, -2)$:

  • Midpoint: $\left( \frac{-6 + (-5)}{2}, \frac{-6 + (-2)}{2} \right) = \left( \frac{-11}{2}, \frac{-8}{2} \right) = (-5.5, -4)$

Application: The midpoint formula is used in geometry, construction, and contextual problems requiring the location of a central point.

Set and Interval Notation

Set Notation

Set notation is used to list unique items or define a rule for a collection of elements. Sets are written using curly braces { }.

• Example: The set of months with birthdays: {January, April, May}

• Example: Multiples of three between 0 and 21: {0, 3, 6, 9, 12, 15, 18, 21}

Interval Notation

Interval notation describes a range of values using parentheses ( ) for values not included and brackets [ ] for values included. The union symbol (U) combines intervals.

• Example: $[2, 7]$ means all values from 2 to 7, inclusive.

• Example: $(-7, -3)$ means all values between -7 and -3, not including endpoints.

• Example: $[60, 120]$ represents all values from 60 to 120, inclusive.

Application: Set and interval notation are used to describe solution sets, domains, and ranges in algebra and precalculus.

Summary Table: Distance and Midpoint Formulas

Concept	Formula	Example
Distance between $(x_1, y_1)$ and $(x_2, y_2)$	$\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$	Between $(-3, -5)$ and $(3, 3)$: $10$
Midpoint between $(x_1, y_1)$ and $(x_2, y_2)$	$\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$	Between $(-6, -6)$ and $(-5, -2)$: $(-5.5, -4)$