91Ó°ÊÓ

The coordinate plane is a two-dimensional surface used for plotting points, lines, and curves. It is divided into four quadrants by a horizontal line called the x-axis and a vertical line called the y-axis. These axes intersect at the origin, which is denoted as \(0,0\).

To locate a point on this plane, you need a pair of coordinates \(x, y\). The x-coordinate tells you how far to move horizontally from the origin (left or right), and the y-coordinate tells you how far to move vertically (up or down).

For example, the point \(-1, 5\)\, means 1 unit to the left along the x-axis and 5 units up along the y-axis from the origin. Conversely, the point \(6, 3\)\ moves 6 units to the right and 3 units up.

• Quadrants: The plane is divided into four quadrants: Top-right (I), Top-left (II), Bottom-left (III), Bottom-right (IV).
• Axis: The x-axis and y-axis are perpendicular.
• Origin: The meeting point of both axes.

Understanding the coordinate plane is essential for graphing and solving geometric problems.

The Pythagorean theorem is a fundamental principle in mathematics that relates the sides of a right triangle. According to this theorem, in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. It is expressed as: \[ a^2 + b^2 = c^2 \]

In order to find the distance between two points on the coordinate plane, we can treat the segment connecting them as the hypotenuse of a right triangle. The differences in the x-values and y-values of the points form the other two sides of that triangle.

By applying the Pythagorean theorem in this context, we can derive the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\)\ with the formula:

• Difference in x-values: \(x_2 - x_1\)
• Difference in y-values: \(y_2 - y_1\)
• The distance formula: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

This is essentially applying the Pythagorean theorem to the distance between two points.

Plotting points on a coordinate plane is a straightforward process. It involves placing the points on the grid based on their x and y coordinates, which tell you the exact location on the plane.

To effectively plot a point such as \(x, y\)\, follow these steps:

• Identify the x-coordinate and move either left or right along the x-axis from the origin.
• Identify the y-coordinate and move either up or down along the y-axis from the current position.
• Mark the point where your movements meet.

For instance, to plot the point \(-1, 5\)\, you move 1 unit left from the origin on the x-axis and then 5 units up on the y-axis. Similarly, for the point \(6, 3\)\, you move 6 units right and 3 units up.

Plotting accurately helps in solving problems involving graphs and distances, making it a foundational skill in geometry.