Coordinate geometry, also known as analytic geometry, is a branch of mathematics that deals with the study of geometric shapes using algebraic methods. It combines the principles of geometry and algebra to analyze and describe the properties and relationships of points, lines, curves, and shapes in a coordinate plane.

In coordinate geometry, points are represented by ordered pairs of numbers called coordinates. The two coordinates, usually denoted as (x, y), represent the horizontal and vertical distances from a reference point, typically the origin (0, 0). The x-coordinate represents the distance along the horizontal axis (usually the x-axis), while the y-coordinate represents the distance along the vertical axis (usually the y-axis).

Formula:

- Distance Formula: The distance between two points (x1, y1) and (x2, y2) in a coordinate plane can be calculated using the distance formula:

d = √((x2 – x1)^2 + (y2 – y1)^2)

Example: Consider two points, A(3, 4) and B(7, 9), in a coordinate plane. We can find the distance between these two points using the distance formula.

Using the distance formula: d = √((7 – 3)^2 + (9 – 4)^2) = √(4^2 + 5^2) = √(16 + 25) = √41

Therefore, the distance between points A(3, 4) and B(7, 9) is √41 units.

- Slope Formula: The slope of a line passing through two points (x1, y1) and (x2, y2) can be calculated using the slope formula:

m = (y2 – y1) / (x2 – x1)

Example: Consider two points, C(2, 5) and D(6, 9), in a coordinate plane. We can find the slope of the line passing through these two points using the slope formula.

Using the slope formula: m = (9 – 5) / (6 – 2) = 4 / 4 = 1

Therefore, the slope of the line passing through points C(2, 5) and D(6, 9) is 1.

These are just two examples of formulas used in coordinate geometry. There are several other formulas and concepts, such as the midpoint formula, the equation of a line, and more, that are integral to the study of coordinate geometry.

The slope of a line is a fundamental concept in coordinate geometry. It represents the measure of the steepness or incline of a line and is denoted by the letter “m.”

Formula: The slope of a line passing through two points (x1, y1) and (x2, y2) can be calculated using the slope formula:

m = (y2 – y1) / (x2 – x1)

Example: Consider two points, A(2, 3) and B(5, 9), in a coordinate plane. We can find the slope of the line passing through these two points using the slope formula.

Using the slope formula: m = (9 – 3) / (5 – 2) = 6 / 3 = 2

Therefore, the slope of the line passing through points A(2, 3) and B(5, 9) is 2.

The slope of a line provides important information about its characteristics. For instance, a positive slope indicates an upward direction from left to right, while a negative slope indicates a downward direction. A slope of zero represents a horizontal line, and an undefined slope corresponds to a vertical line. Slope is also used in determining parallel and perpendicular lines, as lines with the same slope are parallel, while lines with negative reciprocal slopes are perpendicular to each other.

The angle between two lines is a concept in coordinate geometry that measures the angle formed between two lines intersecting at a point. It can be calculated using the slope of the lines.

Formula: The angle θ between two lines with slopes m1 and m2 can be calculated using the following formula:

θ = arctan(|(m1 – m2) / (1 + m1 * m2)|)

Here, arctan denotes the inverse tangent function.

Example: Consider two lines with slopes m1 = 2 and m2 = -1/2. We can find the angle between these two lines using the angle formula.

Using the angle formula: θ = arctan(|(2 – (-1/2)) / (1 + 2 * (-1/2))|) = arctan(|(2 + 1/2) / (1 – 1)|) = arctan(|5/2 / 0|) = arctan(∞) = 90°

Therefore, the angle between the lines with slopes m1 = 2 and m2 = -1/2 is 90 degrees.

It is important to note that the angle between two lines is always measured in degrees or radians, depending on the convention used. In the example above, the angle was measured in degrees. Additionally, the absolute value of the expression inside the arctan function is taken to ensure a positive angle regardless of the slopes’ signs.

Keep in mind that this formula calculates the angle between lines, not the angle of intersection at a specific point.