In our daily lives, we constantly describe locations and positions. When giving directions, we might say 'go 3 blocks north and 2 blocks east' to help someone find a destination. This natural way of describing position using references shows our intuitive understanding of location. In mathematics, we need a systematic and precise way to describe positions on a plane. This leads us to the concept of a coordinate system, which uses two perpendicular reference lines to create a framework for describing any point's location.
To construct a rectangular coordinate system, we start with two perpendicular lines. The horizontal line is called the x-axis, and the vertical line is called the y-axis. These axes intersect at a point called the origin, labeled as (0,0). Each axis has positive and negative directions: on the x-axis, positive values extend to the right and negative values to the left. On the y-axis, positive values go upward and negative values downward. We mark unit lengths along both axes with numbers to create a precise measurement system.
Points in the coordinate system are represented using ordered pairs written as (x, y). The first number, x, represents the horizontal distance from the origin, while the second number, y, represents the vertical distance. Let's plot some examples: the point (3, 2) means 3 units right and 2 units up. The point (-1, 4) means 1 unit left and 4 units up. The point (0, -3) lies on the y-axis, 3 units down. And (-2, -1) means 2 units left and 1 unit down. The order of coordinates is crucial: (3, 2) is completely different from (2, 3).
The coordinate plane is divided into four regions called quadrants by the x and y axes. These quadrants are numbered using Roman numerals. Quadrant I is in the upper right, where both x and y coordinates are positive. Quadrant II is in the upper left, where x is negative and y is positive. Quadrant III is in the lower left, where both coordinates are negative. Quadrant IV is in the lower right, where x is positive and y is negative. Let's see examples: (2,1) is in Quadrant I, (-2,1) is in Quadrant II, (-2,-1) is in Quadrant III, and (2,-1) is in Quadrant IV. Points that lie exactly on the axes, like (3,0) or (0,2), don't belong to any quadrant.
Using coordinates, we can calculate distances between points and find geometric relationships. The distance formula is based on the Pythagorean theorem: d equals the square root of (x2 minus x1) squared plus (y2 minus y1) squared. Let's find the distance between points A(1,2) and B(4,6). We form a right triangle where the horizontal leg is 3 units and the vertical leg is 4 units. Using the formula: d equals square root of (4-1)² plus (6-2)², which equals square root of 9 plus 16, equals 5. We can also find the midpoint between two points using the midpoint formula: M equals the average of the x-coordinates and the average of the y-coordinates, giving us (2.5, 4).