Coordinate Geometry

Introduction

Coordinate geometry is a branch of mathematics that deals with the study of geometry using algebraic methods. In this topic, we use the Cartesian coordinate system to represent points, lines, and shapes on a plane. The Cartesian coordinate system consists of two perpendicular axes, the x-axis, and the y-axis, which intersect at the origin (0, 0).

Coordinates and Points

• Coordinates: Coordinates are pairs of numbers that represent the location of a point in a plane.
• Point: A point is represented by a pair of coordinates (x, y), where x is the distance along the x-axis and y is the distance along the y-axis.

Example: Find the coordinates of point A in the Cartesian plane if A is located at (3, 4). Solution: The coordinates of point A are (3, 4).

Distance Formula

• Distance Formula: The distance between two points A(x1, y1) and B(x2, y2) in a plane is given by: $$ \sqrt{(x2 - x1)^2 + (y2 - y1)^2} $$

Example: Find the distance between points A(2, 3) and B(5, 7). Solution: Using the distance formula: $$ \sqrt{(5 - 2)^2 + (7 - 3)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 $$

Midpoint Formula

• Midpoint Formula: The midpoint of a line segment with endpoints A(x1, y1) and B(x2, y2) is given by: $$ \left(\frac{x1 + x2}{2}, \frac{y1 + y2}{2}\right) $$

Example: Find the midpoint of the line segment with endpoints A(1, 3) and B(7, 9). Solution: Using the midpoint formula: $$ \left(\frac{1 + 7}{2}, \frac{3 + 9}{2}\right) = (4, 6) $$

Slope of a Line

• Slope: The slope of a line passing through two points A(x1, y1) and B(x2, y2) is given by: $$ \frac{y2 - y1}{x2 - x1} $$

Example: Find the slope of the line passing through points A(2, 3) and B(4, 7). Solution: Using the slope formula: $$ \frac{7 - 3}{4 - 2} = \frac{4}{2} = 2 $$

Equation of a Line

• Equation of a Line: The equation of a line passing through a point (x1, y1) with slope m is given by the point-slope form: $$ y - y1 = m(x - x1) $$

Example: Find the equation of the line passing through point (2, 3) with slope 4. Solution: Using the point-slope form: $$ y - 3 = 4(x - 2) $$ $$ y - 3 = 4x - 8 $$ $$ y = 4x - 5 $$

Common Mistakes

• Forgetting to square the differences when calculating distances.
• Misinterpreting the signs when finding the slope of a line.
• Incorrectly applying the midpoint formula by forgetting to divide by 2.

Key Points

• Coordinates are pairs of numbers representing the location of a point.
• The distance between two points is calculated using the distance formula.
• The midpoint of a line segment is found using the midpoint formula.
• The slope of a line is determined by the difference in y-coordinates over the difference in x-coordinates.
• The equation of a line can be found using the point-slope form.

Practice Questions

1. Find the distance between points A(1, 2) and B(5, 8).

Solution: Using the distance formula: $$ \sqrt{(5 - 1)^2 + (8 - 2)^2} = \sqrt{4^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52} = 2\sqrt{13} $$

2. Find the midpoint of the line segment with endpoints C(2, 4) and D(6, 10).

Solution: Using the midpoint formula: $$ \left(\frac{2 + 6}{2}, \frac{4 + 10}{2}\right) = (4, 7) $$

3. Determine the slope of the line passing through points E(3, 5) and F(7, 9).

Solution: Using the slope formula: $$ \frac{9 - 5}{7 - 3} = \frac{4}{4} = 1 $$

4. Find the equation of the line passing through point G(4, 6) with slope -2.

Solution: Using the point-slope form: $$ y - 6 = -2(x - 4) $$ $$ y - 6 = -2x + 8 $$ $$ y = -2x + 14 $$

5. Given points H(0, 2) and I(3, 6), find the distance between them.

Solution: Using the distance formula: $$ \sqrt{(3 - 0)^2 + (6 - 2)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 $$