CONCEPT PREVIEW Fill in the blank(s) to correctly complete each sentence.The midpoint of the segment joining (0, 0)and (4, 4)is ________.

Verified step by step guidance

1. Identify the formula for the midpoint of a line segment, which is given by \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \).
2. Substitute the coordinates of the given points into the formula: \( x_1 = 0, y_1 = 0, x_2 = 4, y_2 = 4 \).
3. Calculate the x-coordinate of the midpoint: \( \frac{0 + 4}{2} \).
4. Calculate the y-coordinate of the midpoint: \( \frac{0 + 4}{2} \).
5. Combine the calculated x and y coordinates to find the midpoint of the segment.

Key Concepts

Midpoint Formula

The midpoint formula is used to find the point that is exactly halfway between two given points in a coordinate plane. It is calculated by averaging the x-coordinates and the y-coordinates of the two points. For points (x1, y1) and (x2, y2), the midpoint M is given by M = ((x1 + x2)/2, (y1 + y2)/2). This concept is essential for solving problems involving line segments in geometry.

Coordinate System

A coordinate system is a two-dimensional plane defined by a horizontal axis (x-axis) and a vertical axis (y-axis). Each point in this system is represented by an ordered pair (x, y), where 'x' indicates the horizontal position and 'y' indicates the vertical position. Understanding how to navigate this system is crucial for accurately determining locations and relationships between points.

Distance Between Points

The distance between two points in a coordinate plane can be calculated using the distance formula, which is derived from the Pythagorean theorem. For points (x1, y1) and (x2, y2), the distance d is given by d = √((x2 - x1)² + (y2 - y1)²). While this concept is not directly needed to find the midpoint, it provides context for understanding the relationship between points and segments in geometry.