Orienting Yourself: The Use of Coordinates · Lesson 2 of 5
Applying Coordinates: The Room Layout
“Latitude tells you how close you are to the equator, while longitude exists primarily to remind you that Greenwich, England decided it was the center of time and space.”
• Understand how a real room can be represented as a scaled two-dimensional map. • Use a fixed reference point and perpendicular directions to describe positions. • Interpret coordinates as distances from reference walls or axes. • Use coordinates to measure doors, furniture and rectangular spaces. • Understand what information a floor plan can and cannot represent.
Introduction
Coordinates are much more than pairs of numbers written inside brackets. They give us a precise way to describe the position of an object on a flat surface.
Imagine designing the layout of a small study room. The room contains a desk, a chair, a bookshelf, a lamp, and a storage cabinet. Saying that the desk is “near the left wall” or that the bookshelf is “a little farther from the door” gives only a rough idea of their positions. Two people could easily interpret such descriptions differently.
A better method is to draw the room on a rectangular grid. We can choose a fixed point as the reference point and use horizontal and vertical lines to locate every object. Each position can then be represented by a pair of numbers called coordinates.
For example, if the centre of a desk is marked at ((4, 3)), the coordinates tell us exactly how far to move horizontally and vertically from the reference point to reach it. The same method can be used to mark the positions of the bookshelf, lamp, or any other object.
A scale can also be used so that the drawing represents the actual room accurately. For instance, one unit on the grid might represent one metre in the real room. In this way, a real space can be converted into a simple mathematical model.
Once objects are represented by points on a coordinate plane, mathematics allows us to do much more than simply locate them. We can calculate the distance between two points, find the midpoint of a line segment, and determine the coordinates of points that divide a segment in a particular ratio.
This is the central idea of coordinate geometry—using numbers, algebra, and geometry together to describe and analyse positions accurately.
A scale tells us how a measured length on a drawing corresponds to a real length. For example, 1 cm : 1 foot means that every 1 cm on the drawing represents 1 foot in the actual room.
Why a Floor Plan Is Two-Dimensional
A floor plan records two directions along the floor: left–right and forward–backward. It does not record height. This is why the corners of a bed or wardrobe touching the floor can be located on the plan, while the exact position of a window above the floor cannot be fully described by the same two numbers.
A two-dimensional coordinate map can describe the footprint of an object, but it cannot by itself tell us the object's height or how high a window is above the floor.
Choosing a Reference Corner
To describe positions consistently, we first choose one fixed point as a reference. Suppose the lower-left corner of the room is chosen as O. The bottom wall can serve as the horizontal reference line and the left wall as the vertical reference line. A point is then described by how far it lies from these two lines.
A point on the bottom wall has no vertical displacement from that wall, so its second coordinate is zero. A point on the left wall has no horizontal displacement from that wall, so its first coordinate is zero. These observations later become the standard rules for points on the coordinate axes.
Reading a Door from Coordinates
Problem
Suppose one end of a room door is D₁ = (7.5, 0) and the other end is R₁ = (11.5, 0). What do these coordinates tell us, and how wide is the door?
- 1.Both points have second coordinate 0, so they lie on the horizontal reference wall.
- 2.D₁ is 7.5 units from the vertical reference wall.
- 3.R₁ is 11.5 units from the vertical reference wall.
- 4.The width is the difference between their horizontal positions.
- 5.Width = 11.5 − 7.5 = 4 units.
- 6.If one unit represents one foot, the door is 4 feet wide.
Comparing Two Openings
Problem
A bathroom door has endpoints B₁ = (0, 1.5) and B₂ = (0, 4). Find its width and compare it with a 4-foot room door.
- 1.Both points have first coordinate 0, so the door lies along the vertical reference wall.
- 2.Its width is the difference between the vertical coordinates.
- 3.Width = 4 − 1.5 = 2.5 units.
- 4.So the bathroom door is 2.5 feet wide.
- 5.It is 1.5 feet narrower than a 4-foot room door.
Coordinates of a Rectangle
Rectangular furniture is easy to describe with coordinates because opposite sides are parallel. If the sides are aligned with the reference directions, points on the same horizontal side share the same second coordinate, and points on the same vertical side share the same first coordinate.
Problem
Three feet of a rectangular table are at (8, 9), (11, 9) and (11, 7), with its sides parallel to the reference directions. Find the fourth foot and the floor dimensions.
- 1.The points (8, 9) and (11, 9) have the same second coordinate, so they form a horizontal side.
- 2.The points (11, 9) and (11, 7) have the same first coordinate, so they form a vertical side.
- 3.The missing corner must have first coordinate 8 and second coordinate 7.
- 4.Therefore the fourth foot is at (8, 7).
- 5.Horizontal length = 11 − 8 = 3 units.
- 6.Vertical width = 9 − 7 = 2 units.
- 7.The floor plan therefore shows a 3 by 2 rectangle. It still does not reveal the table's height.
Planning New Spaces
Coordinates can also be used to place new objects. If we know the required width and length of a washbasin, toilet or dining table, we can choose one corner and generate the remaining corners by moving the required number of units horizontally and vertically.
Problem
A 3 ft × 2 ft washbasin area has lower-left corner (−5, 1), with sides parallel to the reference directions. Find all four corners.
- 1.Start with (−5, 1).
- 2.Move 3 units horizontally: (−2, 1).
- 3.Move 2 units vertically from the starting point: (−5, 3).
- 4.The opposite corner combines the new horizontal and vertical positions: (−2, 3).
- 5.The four corners are (−5,1), (−2,1), (−2,3) and (−5,3).
Coordinates as a Design Tool
A coordinate floor plan can help test whether a door will collide with a wardrobe, whether a table fits in a chosen area, or whether furniture is centred. The important idea is that a coordinate is not just a point label: it is a precise location that can be measured and compared.
Practice Problems
- A rectangular study table has three corners at (6, 8), (10, 8) and (10, 5). Find the fourth corner and its floor dimensions.
- A door lies on the horizontal reference line from (6.5, 0) to (10, 0). Find its width.
- A bathroom door lies on the vertical reference line from (0, 2) to (0, 5). Find its width and compare it with the previous door.
- Mark a 3 × 2 rectangular washbasin space with lower-left corner (−6, 1). Write all four coordinates.
- Explain why the height of a table or the vertical position of a window cannot be determined from a floor plan alone.
Key Takeaways
• A real floor can be represented on a scaled grid. • A fixed corner and two perpendicular directions create a precise location system. • Coordinates can describe positions, dimensions and clearances. • Rectangles can be reconstructed from shared horizontal and vertical coordinates. • A floor plan is two-dimensional, so height information is not included.
Next, we formalise the room-grid idea into the 2-D Cartesian coordinate system, with axes, an origin, quadrants and ordered pairs.