Coordinate Geometry • Topic 1 of 3

Cartesian Plane and Ordered Pairs

What is the Coordinate System? The Cartesian coordinate system (named after mathematician René Descartes) is a method of locating points on a plane using two perpendicular number lines called axes.

The Axes:

  • X-axis: The horizontal number line (left to right)
  • Y-axis: The vertical number line (up and down)
  • Origin: The point where the X-axis and Y-axis intersect, labeled as \(O(0, 0)\)

What are Ordered Pairs? An ordered pair \((x, y)\) represents the coordinates of a point, where:

  • \(x\) = the distance from the Y-axis (called the x-coordinate or abscissa)
  • \(y\) = the distance from the X-axis (called the y-coordinate or ordinate)

Important Rules:

  • The order matters! \((2, 3)\) is different from \((3, 2)\)
  • The first number always refers to the X-axis (horizontal position)
  • The second number always refers to the Y-axis (vertical position)

How to Plot a Point:

  • Start at the origin \((0, 0)\)
  • Move horizontally along the X-axis according to \(x\):
  • Positive \(x\) → move RIGHT
  • Negative \(x\) → move LEFT
  • Move vertically according to \(y\):
  • Positive \(y\) → move UP
  • Negative \(y\) → move DOWN
THE CARTESIAN PLANE X Y 0 1 2 3 4 -1 -2 -3 1 2 3 -1 -2 -3 Q I (+,+) Q II (-,+) Q III (-,-) Q IV (+,-) A(3,2) B(-2,2) C(-2,-2) D(3,-2)
Solved Examples
1
Worked Example
Write the coordinates of the points: A point that is 5 units right of origin and 3 units up.
Solution- Right of origin → positive x-coordinate: \(x = 5\) - Up from origin → positive y-coordinate: \(y = 3\) - **Answer:** \((5, 3)\) *Example 2: Plot the point \((-3, 4)\) on the coordinate plane. Solution: - Start at \((0, 0)\) - Move 3 units LEFT (because \(x = -3\) is negative) - Move 4 units UP (because \(y = 4\) is positive) - Mark the point and label it \(P(-3, 4)\) - **Answer:** Point plotted in Quadrant II *Example 3: A rectangle has vertices at \((1, 1)\), \((5, 1)\), \((5, 4)\), and \((1, 4)\). Find the length and width. Solution: - Length = distance along X-axis = \(5 - 1 = 4\) units - Width = distance along Y-axis = \(4 - 1 = 3\) units - **Answer:** Length = 4 units, Width = 3 units

Key Points

  • The Cartesian plane has X-axis (horizontal) and Y-axis (vertical)
  • Origin is \((0, 0)\) where axes intersect
  • Ordered pair \((x, y)\): first is x-coordinate, second is y-coordinate
  • Positive x → right; negative x → left
  • Positive y → up; negative y → down
  • The order of numbers in a pair is extremely important
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Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.A point is written as an ordered pair:
Explanation: $(x,y)$, $x$ first.
Q2.The origin has coordinates:
Explanation: $(0,0)$.
Q3.The horizontal axis is the:
Explanation: $x$-axis.
Q4.The $x$-coordinate is also called the:
Explanation: Abscissa.
Practice more MCQs → 📝 Assignment · 35 marks