Section 2.1: The Distance and Midpoint Formulas

nCartesian (Rectangular) Coordinate System

The Cartesian coordinate system is a two-dimensional plane defined by a horizontal axis (x-axis) and a vertical axis (y-axis). Points are located using ordered pairs (x, y), where x represents the horizontal position and y represents the vertical position.

Distance Formula

The distance formula is used to find the distance between two points in the Cartesian plane. If the points are and , the formula is derived from the Pythagorean Theorem:

• Formula:

• Application: Useful for measuring straight-line distances between any two points.

Example 1: Find the distance between (0, 1) and (3, 5):

Using the Distance Formula to Prove Geometric Properties

The distance formula can be used to prove geometric properties, such as whether three points form a right triangle.

• Pythagorean Theorem:

• Example 2: Given points A = (-2, 1), B = (2, 3), and C = (3, 1): - Find the distances AB, BC, and AC. - Check if the triangle formed is a right triangle using the Pythagorean Theorem.

• Area of Triangle:

Example: Calculate the area using the base and height found from the coordinates.

Finding Points at a Specific Distance

Sometimes, you are asked to find all points with a given x-coordinate that are a specific distance from another point.

• Example 3: Find all points with x = 3 whose distance from (-2, -1) is 13 units. - Set up the equation: - Solve for y.

Midpoint Formula

The midpoint formula finds the point exactly halfway between two given points and :

• Formula:

• Application: Useful for bisecting line segments or finding centers.

Example 4: Find the midpoint between and :

Finding an Endpoint Given the Midpoint

If you know one endpoint and the midpoint, you can find the other endpoint by solving the midpoint formula for the unknown coordinates.

• Example 5: The midpoint is (-1, 4), and . Find : - Set up equations: and - Solve for x and y.

Summary Table: Distance and Midpoint Formulas