Coordinate Geometry, also known as Cartesian Geometry, is a branch of mathematics that deals with representing points and shapes using a numerical system. This chapter introduces you to the fundamental concepts and formulas used to analyze geometric problems on a two-dimensional plane.
Distance Formula
This formula calculates the distance between two points on the coordinate plane, denoted by P1(x1, y1) and P2(x2, y2).
Formula:
Distance (d) = √((x2 – x1)² + (y2 – y1)²)
Explanation:
Imagine the points P1 and P2 on a graph. The distance between them forms the hypotenuse of a right triangle. The difference in their x-coordinates (x2 – x1) represents the change along the x-axis, and the difference in their y-coordinates (y2 – y1) represents the change along the y-axis. By applying the Pythagorean Theorem, we find the square root of the sum of the squared differences to determine the distance.
Examples:
1. Find the distance between points A(2, 3) and B(5, 1).
Solution:
d = √((5 – 2)² + (1 – 3)²)
= √((3)² + (-2)²)
= √(13)
The distance between A and B is √(13) units.
2. Two points, P(-1, 4) and Q(3, -2), are plotted on a graph. What is the distance between them?
Solution:
d = √((3 – (-1))² + (-2 – 4)²)
= √((4)² + (-6)²)
= √(52)
= 2√13
The distance between P and Q is 2√13 units.
3. If point C is at (0, 0) and D is at (6, 8), find the distance between them.
Solution:
d = √((6 – 0)² + (8 – 0)²)
= √((6)² + (8)²)
= √(100)
= 10
The distance between C and D is 10 units.
Section Formula
This formula helps us find the coordinates of a point that divides a line segment joining two given points in a specific ratio. Let P1(x1, y1) and P2(x2, y2) be the two points, and the ratio of the division is m:n (m + n ≠ 0).
Formula:
Coordinates of dividing point = ((mx2 + nx1) / (m + n), (my2 + ny1) / (m + n))
Explanation:
Imagine a line segment connecting P1 and P2. The section formula allows us to locate a point on this line segment that separates it into two parts with a specific length ratio of m:n. The formula essentially calculates the weighted average of the x and y coordinates of the two points based on the given ratio.
Examples:
1. Find the coordinates of the point that divides the line segment joining A(2, 5) and B(8, 1) in the ratio 2:3.
Solution:
Dividing point = ((2 * 8 + 3 * 2) / (2 + 3), (2 * 1 + 3 * 5) / (2 + 3))
= (16 + 6) / 5, (2 + 15) / 5
= 22/5, 17/5
The point dividing the line segment in the ratio 2:3 is (22/5, 17/5).
2. If point P(3, 4) is one-third of the way from point Q(-1, 2) to point R (a, b), what are the coordinates of R?
Solution:
In this case, the ratio is 1:2 (since P is one-third of the way from Q).
3 = ((1 * a) + (2 * -1)) / (1 + 2)
a = 7
4 = ((1 * b) + (2 * 2)) / (1 + 2)
b = 0
Therefore, the coordinates of R are (7, 0).
Midpoint Formula
This formula determines the coordinates of the midpoint of a line segment connecting two points. Let P1(x1, y1) and P2(x2, y2) represent the two points.
Formula:
Midpoint coordinates = ((x1 + x2) / 2, (y1 + y2) / 2)
Explanation:
The midpoint of a line segment essentially divides it into two segments of equal length. This formula calculates the average of the x-coordinates and the average of the y-coordinates of the two points, resulting in the coordinates of the midpoint.
Example:
1. Let S be the midpoint between points X(-2, 5) and Y(4, -1). Find the coordinates of S.
Solution:
Midpoint S = ((-2 + 4) / 2, (5 – 1) / 2)
= (2 / 2, 4 / 2)
= (1, 2)
Therefore, the midpoint S has coordinates (1, 2).
Slope Formula
The slope formula calculates the steepness or inclination of a non-vertical line in the coordinate plane. It represents the rise (change in y) over the run (change in x) between two points on the line. Let P1(x1, y1) and P2(x2, y2) be the two points.
Formula:
Slope (m) = (y2 – y1) / (x2 – x1)
Explanation:
Imagine a line connecting P1 and P2. The slope signifies how much the y-coordinate changes (rises or falls) for every unit change in the x-coordinate as you move along the line. A positive slope indicates an upward trend from left to right, while a negative slope indicates a downward trend. A slope of 0 represents a horizontal line, and the slope is undefined for a vertical line (infinite rise over zero run).
Examples:
1. Calculate the slope of the line passing through points A(1, 3) and B(4, 7).
Solution:
m = (7 – 3) / (4 – 1)
= 4 / 3
The slope of the line is 4/3, indicating an upward trend from left to right with a rise of 4 units for every 3 units moved to the right.
2. Find the slope of the line connecting points C(-2, -1) and D(-2, 5).
Solution:
m = (5 – (-1)) / (-2 – (-2)) // Since x-coordinates are the same, the change in x is 0
= 6 / 0
The slope is undefined in this case because the line is vertical (no change in x-coordinate).
3. If line E has a slope of -1/2 and passes through point F(3, -4), what are the possible coordinates of another point on this line?
Solution:
Since the slope is -1/2, for every 1 unit you move down (negative y-coordinate change), you move 2 units to the right (positive x-coordinate change). Let’s consider two possibilities:
- Move 1 unit down and 2 units to the right from F(3, -4): New point G(3 + 2, -4 – 1) = (5, -5)
- Move 2 units down and 4 units to the right from F(3, -4): New point H(3 + 4, -4 – 2) = (7, -6)
Both points G(5, -5) and H(7, -6) are valid options as they maintain the slope of -1/2.
Equation of a Line
There are two main ways to represent the equation of a line in coordinate geometry:
- Slope-Intercept Form (y = mx + c):
- This is the most common form, where:
- m represents the slope (as defined earlier).
- c represents the y-intercept (the point where the line crosses the y-axis).
- Notes:
- A steeper slope (larger absolute value of m) indicates a more pronounced slant.
- A positive c-value results in the y-intercept being above the origin, while a negative c-value places it below.
- This is the most common form, where:
- Point-Slope Form (y – y1 = m(x – x1)):
- This form is useful when you know the slope (m) and the coordinates of one point on the line (x1, y1).
- Notes:
- This form emphasizes the change in y relative to a specific point.
Example:
Write the equation of the line that passes through point P(2, 1) and has a slope of -2.
Solution (using slope-intercept form):
m = -2, x1 = 2, y1 = 1 y = (-2)x + c (substitute m and x1, y1)
To find c (y-intercept), we can plug the known point (2, 1) into the equation:
1 = (-2)(2) + c 1 = -4 + c c = 5
Therefore, the equation of the line is y = -2x + 5.
Distance Formula between a Point and a Line
This formula calculates the shortest distance between a point (P) and a line that is not vertical. The line is represented by its equation in slope-intercept form (y = mx + c).
Formula:
Distance (d) = | (y2 – y1) – m(x2 – x1) | / √(m² + 1)
- (x1, y1) are the coordinates of point P.
- m and c define the slope and y-intercept of the line.
Notes:
- The absolute value ensures the distance is non-negative.
- The denominator term (√(m² + 1)) represents the distance between a point on the line and a point directly above or below it (perpendicular distance).
Area of a Triangle
The area of a triangle can be calculated using its coordinates in several ways, depending on the information available. Here’s a common formula:
Formula:
Area = 1/2 * | (x1 * y2 – x2 * y1) + (x2 * y3 – x3 * y2) + (x3 * y1 – x1 * y3) |
- (x1, y1), (x2, y2), and (x3, y3) are the coordinates of the three vertices of the triangle.
Notes:
- This formula works for any triangle, regardless of its orientation.
- The absolute value ensures the area is positive.
Applications of Coordinate Geometry
Coordinate Geometry (or Cartesian Geometry) finds applications in various fields beyond pure mathematics. Here are some key areas where it plays a crucial role:
- Computer Graphics:
- Coordinate systems are fundamental for representing and manipulating images, shapes, and objects on computer screens. Pixels on a display are essentially addressed using coordinates, allowing for precise control of visual elements.
- Animation and Game Design:
- Animators and game developers heavily rely on coordinate geometry to define the movement and positions of characters, objects, and backgrounds within a virtual environment.
- Navigation and GPS Systems:
- Global Positioning Systems (GPS) utilize coordinates (latitude and longitude) to pinpoint your location on Earth. By referencing these coordinates with digital maps stored in navigation systems, you can determine routes and track your position.
- Robotics and Automation:
- Robots and automated machines rely on coordinate systems to precisely control their movements and interactions with the surrounding environment. This allows for accurate positioning of robotic arms, path planning for autonomous vehicles, and object manipulation in industrial settings.
- Physics and Engineering:
- Coordinate geometry is a cornerstone for analyzing motion, forces, and other physical phenomena. It helps engineers model structures, design machines, and simulate different scenarios using mathematical representations.
- Architecture and Urban Planning:
- Architects and urban planners utilize coordinate systems to design buildings, lay out city infrastructure, and optimize land use. This ensures precise measurements and facilitates the visualization of planned structures.
Summary of Coordinate Geometry
Coordinate Geometry bridges the gap between algebra and geometry by providing a numerical framework for representing geometric shapes. Key concepts include:
- Coordinate System: A system of axes (usually x and y) used to represent points using ordered pairs of numbers (coordinates).
- Distance Formulas: Calculating distances between points and the distance between a point and a line.
- Equation of a Line: Representing a line algebraically using slope-intercept form (y = mx + c) or point-slope form.
- Midpoint Formula: Finding the coordinates of the midpoint of a line segment joining two points.
- Area Formulas: Calculate the area of shapes like triangles using their coordinates.
By mastering these core concepts and their applications, you gain a powerful tool for analyzing and solving geometric problems in various real-world contexts.
