Coordinate Number Line

A straight line on which every point corresponds to a number is called the coordinate number line. This line serves as a tool to represent numbers geometrically on a plane. On the coordinate number line, each point is associated with a specific real number in a linear arrangement. For instance, if point AAA is associated with a real number xxx, it is expressed as A(x).

The coordinate number line demonstrates the infinite nature of real numbers. Between any two real numbers, there are infinitely many other real numbers.

Analytic Plane

A system formed by two perpendicular lines intersecting at the same starting point on a plane is called a coordinate system. The intersection point of these lines, referred to as “O,” is called the origin or the starting point. The plane defined around this origin and system is known as the analytic plane.

In the analytic plane, these lines are called axes and are known as coordinate axes. The horizontal line is referred to as the x-axis, while the vertical line is called the y-axis. These two axes divide the analytic plane into four distinct regions, each characterized by the signs of the coordinates of the points within them.

Abscissa and Ordinate

In the coordinate system, the horizontal distance of a point from the x-axis is called the abscissa, and the vertical distance from the y-axis is called the ordinate.

The coordinates of a point are written as (x,y), where x represents the abscissa, and y represents the ordinate. For example, in the point A(3,2), the abscissa is 3, and the ordinate is 2.

Distance of a Point from the Origin

The distance of a point from the origin is the length of the straight line connecting the point to the origin. This distance is determined by the combination of the point’s horizontal and vertical positions.

Distance Between Two Points

In the analytic plane, the distance between two points is the length of the straight line segment connecting them. The distance between these two points is calculated using the following formula:

Coordinates of the Midpoint of a Line Segment

The midpoint of a line segment is the point located in the middle of the two endpoints. This formula determines the position of the midpoint by averaging the coordinates of the two points:

Slope Angle and Slope of a Line

The angle formed by a line with the positive direction of the x-axis is called the slope angle. The tangent of this angle is referred to as the slope of the line. The slope is usually represented by the letter m.

The tangent of acute angles is positive, while the tangent of obtuse angles is negative.

Slope of a Line with Two Known Points

The slope of a line is defined as the ratio of the vertical and horizontal distances between two points on the line. If a line passes through two specific points, the slope between these points represents the ratio of the line’s vertical and horizontal movements.

Open Line Equation and Slope

When the equation of a line is given in the form y = ax + b, it is called the “open line equation.” This form is especially useful for calculating the slope of the line.

Example:

If the equation is y = −2x + 5, so the slope of the line m=−2. This means that for every one-unit increase in x, y decreases by 2 units.

Closed Line Equation and Slope

A Point on the Line

Equation of a Line Given Its Slope and a Point

A Line Intersecting the X-Axis and Y-Axis

In analytical geometry, the equation of a line that intersects the x-axis at point (a,0)(a, 0)(a,0) and the y-axis at point (0,b)(0, b)(0,b) can be derived using the intercept form of a line. This form is particularly useful when both intercepts are known.

Points with Equal Abscissas and Ordinates: y = x

In the Cartesian coordinate system, the points where the x-coordinate and y-coordinate are equal form the line “y = x”.

On this line, the x and y values of each point are equal. For example, points like (1,1), (2,2), and so on lie on this line.

• Slope: 1, because for every 1 unit of horizontal movement, there is 1 unit of vertical movement.
• Passes through the origin: The line y=xy = xy=x passes through the origin (0,0).

Points with Opposite Abscissas and Ordinates

In the Cartesian coordinate system, points where the abscissa (x) and ordinate (y) values have opposite signs form the line y = -x.