Graphs

The Distance and Midpoint Formulas

This section introduces two fundamental tools for analyzing points and line segments in the coordinate plane: the Distance Formula and the Midpoint Formula. These formulas are essential for solving geometric problems algebraically and are foundational in precalculus and analytic geometry.

Distance Formula

The Distance Formula allows us to calculate the distance between any two points in the coordinate plane. It is derived from the Pythagorean Theorem by considering the horizontal and vertical distances between the points as the legs of a right triangle.

• Definition: The distance d between points and is given by:

• Application: Used to find the length of a line segment, the sides of polygons, or to determine if points are collinear or form specific geometric shapes.

Example: Finding the Distance Between Two Points

Find the distance between the points (0, 2) and (4, 5).

• Step 1: Plot the points and connect them with a straight line.

• Step 2: Draw a horizontal and a vertical line to form a right triangle. The horizontal leg has length 4, and the vertical leg has length 3.

• Step 3: Apply the Distance Formula:

• Conclusion: The distance between (0, 2) and (4, 5) is 5 units.

Using the Distance Formula in Geometry

The Distance Formula can be used to solve more complex geometry problems, such as determining the type of triangle formed by three points.

• Given points A, B, and C, calculate the lengths of all sides using the Distance Formula.

• If the sum of the squares of two sides equals the square of the third side, the triangle is a right triangle (by the converse of the Pythagorean Theorem).

• Area of a Right Triangle: If the right angle is at vertex A, the sides AC and AB form the base and height. The area is:

Midpoint Formula

The Midpoint Formula finds the point exactly halfway between two given points in the coordinate plane. This is useful for bisecting line segments or finding centers of geometric figures.

• Definition: The midpoint of the segment from to is:

• Application: Used to find the center of a segment, the centroid of a triangle, or to divide a segment into equal parts.

Example: Finding the Midpoint of a Line Segment

Find the midpoint of the segment joining and .

• Apply the Midpoint Formula:

• Conclusion: The midpoint is at (0, 3).