What this chapter is about
This chapter introduces the coordinate geometry system, a powerful method that connects algebra with geometry. You learn how to describe the exact position of any point on a flat surface using two numbers, called coordinates. This idea was developed by the French mathematician René Descartes, which is why the system is called the Cartesian coordinate system.
In earlier classes, you located places using directions like "3 km north and 2 km east". Now you formalise this idea mathematically. You study how two perpendicular number lines (called axes) create a plane where every point has a unique address written as an ordered pair (x, y).
After studying this chapter, you should be able to plot points on a coordinate plane, identify the coordinates of given points, understand which quadrant a point lies in, and describe positions precisely using mathematical language. This skill forms the foundation for graphing linear equations, which you will study later this year.
Key ideas
- The coordinate plane consists of two perpendicular number lines: the horizontal x-axis and the vertical y-axis. They intersect at a point called the origin, denoted by O.
- The origin has coordinates (0, 0). Every other point on the plane is located by measuring its horizontal and vertical distances from the origin.
- An ordered pair (x, y) gives the coordinates of a point. The first number x is called the abscissa (horizontal distance), and the second number y is called the ordinate (vertical distance). The order matters: (3, 5) and (5, 3) are different points.
- The two axes divide the plane into four regions called quadrants. In Quadrant I, both x and y are positive. In Quadrant II, x is negative and y is positive. In Quadrant III, both are negative. In Quadrant IV, x is positive and y is negative.
- Points lying on the x-axis have y-coordinate equal to zero, written as (a, 0). Points lying on the y-axis have x-coordinate equal to zero, written as (0, b).
- The coordinate system allows us to represent geometric shapes algebraically and solve geometry problems using equations.
Formulas and facts to remember
- Origin: The point (0, 0) where the x-axis and y-axis meet.
- Abscissa: The x-coordinate of a point; it tells the horizontal position (positive to the right, negative to the left).
- Ordinate: The y-coordinate of a point; it tells the vertical position (positive upward, negative downward).
- Quadrant signs: I (+, +), II (−, +), III (−, −), IV (+, −).
- Points on x-axis: Have the form (a, 0) for any real number a.
- Points on y-axis: Have the form (0, b) for any real number b.
- Distance of point (x, y) from x-axis: |y| units.
- Distance of point (x, y) from y-axis: |x| units.
Worked examples
Example 1: Plot the point P(4, 3) on a coordinate plane and state which quadrant it lies in.
Solution: Step 1: Draw the x-axis (horizontal) and y-axis (vertical) intersecting at origin O. Step 2: From the origin, move 4 units to the right along the x-axis (since x = 4 is positive). Step 3: From that position, move 3 units upward parallel to the y-axis (since y = 3 is positive). Step 4: Mark this point as P. Since both coordinates are positive, point P lies in Quadrant I.
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Example 2: Write the coordinates of a point Q that is 5 units to the left of the origin and 2 units below the x-axis.
Solution: Step 1: Moving left from the origin means the x-coordinate is negative. So x = −5. Step 2: Moving below the x-axis means the y-coordinate is negative. So y = −2. Step 3: The coordinates of Q are (−5, −2). This point lies in Quadrant III.
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Example 3: A point R lies on the y-axis, 7 units above the origin. What are its coordinates?
Solution: Step 1: Any point on the y-axis has x-coordinate equal to 0. Step 2: The point is 7 units above the origin, so y = 7. Step 3: The coordinates of R are (0, 7). Since R is on the y-axis, it does not belong to any quadrant.
Common mistakes
- Writing coordinates in the wrong order, such as writing (y, x) instead of (x, y) → Always remember: x comes first, then y. Think "across before up".
- Confusing the signs when plotting in different quadrants → Use the rule: right is positive x, left is negative x; up is positive y, down is negative y.
- Thinking (0, 5) and (5, 0) are the same point → (0, 5) is on the y-axis, while (5, 0) is on the x-axis. They are completely different locations.
- Saying a point on an axis belongs to a quadrant → Points on the axes are not in any quadrant; quadrants are the four open regions between the axes.
- Measuring the ordinate along the x-axis → The ordinate (y-value) is always measured parallel to the y-axis, not along the x-axis.
Quick revision
- The coordinate plane has two axes meeting at the origin (0, 0).
- Every point is written as (x, y), where x is the abscissa and y is the ordinate.
- Quadrant signs: I (+, +), II (−, +), III (−, −), IV (+, −).
- Points on the x-axis have y = 0; points on the y-axis have x = 0.
- The order in an ordered pair matters: (3, 7) ≠ (7, 3).