Chapter 2: Graphs

2.1 The Distance and Midpoint Formulas

This section introduces the foundational concepts of the coordinate plane, focusing on the distance and midpoint formulas. These tools are essential for analyzing geometric relationships between points in two dimensions.

Rectangular (Cartesian) Coordinate System

• Definition: The rectangular coordinate system (or Cartesian plane) consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical), intersecting at the origin (0,0).

• Coordinates: Each point in the plane is represented as an ordered pair (x, y).

• Quadrants: The plane is divided into four quadrants, each with distinct sign combinations for x and y.

Plotting Points

• To plot a point (x, y), move x units along the x-axis and y units along the y-axis.

• Examples:

  • (6, 4): 6 units right, 4 units up

  • (-3, -5): 3 units left, 5 units down

  • (-6, 0): 6 units left, on the x-axis

  • (0, 7): On the y-axis, 7 units up

The Distance Formula

The distance formula calculates the straight-line distance between two points in the plane.

• Formula:

• Derivation: This formula is derived from the Pythagorean Theorem, treating the difference in x and y as the legs of a right triangle.

Example: Finding Distance

• Problem: Find the distance between (1, 3) and (5, 6).

• Solution:

  • Apply the formula:

Example: Distance Between (-4, 5) and (3, 2)

• Solution:

The Midpoint Formula

The midpoint formula finds the point exactly halfway between two given points in the plane.

• Formula:

• Interpretation: The midpoint's coordinates are the averages of the corresponding coordinates of the endpoints.

Example: Finding a Midpoint

• Problem: Find the midpoint between and .

• Solution:

Applications: Triangles in the Coordinate Plane

• Problem: Given points , , and :

  • Plot the points and form triangle ABC.

  • Find the length of each side using the distance formula.

  • Show that the triangle is a right triangle (by checking if the side lengths satisfy the Pythagorean Theorem).

  • Find the area of the triangle (using the formula or the determinant method).

Example: Circle from Diameter Endpoints

• Problem: The endpoints of a diameter are and . Find the radius and the center.

• Solution:

  • Center: Use the midpoint formula:

  • Radius: Half the distance between A and B:

Additional info: The area of a triangle given vertices , , can also be found using the formula: