Section F.1: The Distance and Midpoint Formulas

Plotting Points in the Coordinate Plane

The coordinate plane is a fundamental concept in precalculus, allowing us to visually represent points, lines, and geometric figures. It is formed by two perpendicular number lines that intersect at a point called the origin. The horizontal axis is known as the x-axis, and the vertical axis is the y-axis. These axes divide the plane into four regions called quadrants.

• Ordered Pair (x, y): Each point in the plane is represented by an ordered pair (x, y).

• x-coordinate: The first value in the pair, indicating horizontal position.

• y-coordinate: The second value in the pair, indicating vertical position.

Example: Graph and label the points (−3, 5), (3, 4), (−4, −2), and (0, 4) on the coordinate plane.

The Distance Formula

The distance formula allows us to calculate the straight-line distance between any two points in the coordinate plane. If the points are (x1, y1) and (x2, y2), the distance d between them is given by:

• This formula is derived from the Pythagorean Theorem.

• It is used to find the length of a segment connecting two points.

Example: Find the distance between the points (−2, 2) and (−5, 8).

Applications: Triangles in the Coordinate Plane

By plotting three points, we can form a triangle and use the distance formula to find the lengths of its sides. This is useful for verifying properties such as whether a triangle is a right triangle and for finding its area.

• Example: Consider points A = (−6, 11), B = (3, 9), and C = (1, 0). Plot each point and form triangle ABC.

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

• To verify if the triangle is a right triangle, check if the side lengths satisfy the Pythagorean Theorem.

• To find the area, use the formula for the area of a triangle given coordinates:

The Midpoint Formula

The midpoint formula finds the point exactly halfway between two given points in the coordinate plane. If the endpoints are (x1, y1) and (x2, y2), the midpoint M is:

• This formula is useful for dividing a segment into two equal parts or finding the center of a line segment.

Example: Find the midpoint of the segment whose endpoints are (−4, −2) and (2, 5).

Application: Real-World Example

The midpoint formula can be used in real-world contexts, such as estimating values that change linearly over time. For example, if the poverty threshold for a family of four was $15,604 in 1995 and $19,716 in 2005, the estimated threshold in 2000 (the midpoint year) is:

This represents the estimated poverty threshold in 2000, assuming a straight-line increase.