Orienting Yourself: The Use of Coordinates

1. Chapter at a glance

• A system of coordinates uses two perpendicular lines (x-axis horizontal, y-axis vertical) intersecting at the origin O(0,0) to locate any point in the 2-D plane.
• The coordinate axes divide the plane into four quadrants with fixed sign conventions: I(+,+), II(−,+), III(−,−), IV(+,−).
• The x-coordinate of a point is its perpendicular distance from the y-axis; the y-coordinate is its perpendicular distance from the x-axis.
• Points on the x-axis have coordinates of the form (x,0); points on the y-axis have coordinates of the form (0,y).
• The distance between two points on the same horizontal or vertical line is the absolute difference of the corresponding coordinates.
• For any two points, the Baudhāyana–Pythagoras theorem gives the straight-line distance in the coordinate plane.
• Reflection of a figure across either axis preserves all distances.

2. Definitions, theorems and results

• Origin: The point of intersection of the x-axis and y-axis; its coordinates are (0,0).
• x-axis: The horizontal coordinate axis.
• y-axis: The vertical coordinate axis.
• Cartesian plane (coordinate plane / xy-plane): The plane containing the two perpendicular coordinate axes.
• Quadrants: The four regions into which the axes divide the plane, numbered I–IV anticlockwise.
• Coordinates of a point P: An ordered pair (x,y) where x is the perpendicular distance of P from the y-axis and y is the perpendicular distance of P from the x-axis.
• Points on the axes: Any point on the x-axis is of the form (x,0); any point on the y-axis is of the form (0,y).
• Baudhāyana–Pythagoras theorem (used for distance): In a right-angled triangle the square of the hypotenuse equals the sum of the squares of the other two sides. (Requires proof in the exam when deriving the general distance formula.)
• Distance on axes: The distance between (x₁,y) and (x₂,y) is |x₂−x₁|; the distance between (x,y₁) and (x,y₂) is |y₂−y₁|.
• General distance: The distance between (x₁,y₁) and (x₂,y₂) is √[(x₂−x₁)²+(y₂−y₁)²].
• Reflection property: Reflection across either axis preserves lengths of segments.

3. Formula sheet

4. Solved-example patterns

• Locating points / reading coordinates: Identify the perpendicular distances from each axis; write the ordered pair (x,y) and state the quadrant.
• Placing objects on the grid: Fix three vertices of a rectangle; use equal x- or y-coordinates to find the fourth vertex.
• Distance on or parallel to axes: Subtract the differing coordinate and take absolute value.
• General distance between two points: Form a right triangle with sides parallel to the axes, apply Baudhāyana–Pythagoras theorem (or the derived formula) and simplify.
• Checking collinearity or midpoint relations: Compare distances from a suspected midpoint or verify that the sum of two segments equals the third (using the distance formula).
• Reflection / image points: Change the sign of the appropriate coordinate while keeping the other unchanged; verify distances remain equal.

5. Common mistakes and exam pitfalls

• Writing (y,x) instead of (x,y) or swapping the order of coordinates.
• Forgetting that x-coordinate measures distance from the y-axis (not the x-axis).
• Omitting the absolute-value sign when calculating horizontal or vertical distances.
• Using signed differences instead of absolute values for distances on the axes, leading to negative lengths.
• Assuming all points with one zero coordinate lie in a specific quadrant (they lie on the axes, not in any quadrant).
• Neglecting to check whether a point lies inside, on or outside a circle when radius comparisons are required.
• Missing that reflection preserves distances but changes signs of coordinates.