Floor plans, street grids, screens and circles. Widths come from subtracting coordinates, overlap of circles from comparing a distance with the sum of the radii, and collinearity from adding distances.
Place the origin at one corner of a room and measure along the two walls that meet there. Every object in the room now has coordinates. A rectangular wardrobe is fixed by two opposite corners. A door is a segment along a wall.
Every point inside the room is then to the right of the origin and above it. So a floor plan uses only the first quadrant. This is a choice of where to put the origin, made so that all the numbers are positive. A different choice would give the same room with different numbers.
With coordinates, size questions become subtraction. The width of a door is the difference between the coordinates of its two ends. Whether a wardrobe fits along a wall is a comparison of two differences.
🛡 Plan a room › Move furniture around a 12 ft by 10 ft floor plan, read the corner coordinates, and see when an item blocks the door’s swing.A room door runs from D (8, 0) to R (11.5, 0). A bathroom door runs from B₁ (0, 1.5) to B₂ (0, 4). A wheelchair needs a clear width of about 2.7 ft (32 inches). Does each door allow it?
Room door: both ends have y = 0, so subtract the x-coordinates. 11.5 − 8 = 3.5 ft. That is more than 2.7 ft.
Bathroom door: both ends have x = 0, so subtract the y-coordinates. 4 − 1.5 = 2.5 ft. That is less than 2.7 ft, so it does not allow a wheelchair.
Which coordinate to subtract depends on which way the door runs. A door along the x-axis changes only in x. A door along the y-axis changes only in y.
Two main roads cross at a town centre, one running north–south and one east–west. Every other street is parallel to one of them, 200 m apart.
Name each crossing by the ordered pair (north–south street number, east–west street number). The crossing of the 2nd north–south street and the 5th east–west street is (2, 5).
The order matters here as it did in session 3. (4, 3) and (3, 4) are different crossings. Swapping the numbers puts you at a crossing 200 m east and 200 m south, or the reverse, about 283 m away in a straight line.
A screen 800 pixels wide and 600 pixels high can be given coordinates with x from 0 to 800 and y from 0 to 600. Here the origin is at the bottom-left, as in maths. Many computer graphics systems put the origin at the top-left and count y downwards; the method below is the same either way.
A circular icon of radius 80 is centred at A (100, 150). Does any part of it go off the screen? Compare the centre with each edge:
A second icon of radius 100 is centred at B (250, 230). Do the two circles overlap? Two circles overlap when the distance between their centres is less than the sum of their radii.
AB = √(150² + 80²) = √28,900 = 170
The radii add to 80 + 100 = 180. Because 170 < 180, the circles overlap. A distance greater than 180 would mean they are separate. A distance of exactly 180 would mean they touch at one point.
One comparison decides overlap. If the centres are closer than the sum of the radii, the circles overlap. Computer games use this test to detect collisions between round objects.
Three points are collinear if one straight line passes through all of them. Plotting them and looking is unreliable when one point is only slightly off the line. Distances give an exact test.
If B lies on the segment from A to C, the path A → B → C is the same length as the straight path from A to C. So AB + BC = AC. If B is off the line by any amount, the path through B is longer, and AB + BC > AC.
Take M (−3, −4), A (0, 0) and G (6, 8). MA = √(9 + 16) = 5. AG = √(36 + 64) = 10. MG = √(81 + 144) = √225 = 15. Since 5 + 10 = 15 exactly, the three points are collinear.
🔗 Test it by dragging › Move the middle point one unit off the line and watch AB + BC become larger than AC. The area of the triangle the three points make appears at the same time, and returns to 0 when they line up.Are R (−5, −1), B (−2, −5) and C (4, −12) collinear?
RB = √(3² + (−4)²) = √25 = 5
BC = √(6² + (−7)²) = √85 ≈ 9.220
RC = √(9² + (−11)²) = √202 ≈ 14.213
RB + BC ≈ 14.220, which is greater than RC ≈ 14.213. The sum is larger, so the points are not collinear.
The difference is about 0.007 units. On a plot, the three points would look straight. The calculation shows what the drawing cannot.
A point in the plane is fixed by two numbers. From the signs, you know its quadrant. From the coordinate differences, you can find the distance to any other point, the midpoint, the trisection points, and whether three points are collinear. None of these needs a drawing.
In Class 10 the same coordinates are used to describe whole lines and curves by equations.
1A door runs from (8, 0) to (11.5, 0) along a wall. How wide is it?
2Why does a floor plan normally use only the first quadrant?
3In a street grid, crossings are named (north–south street, east–west street). How many crossings are called (4, 3)?
4A circular icon of radius 80 pixels is centred at (100, 150). Its leftmost point has x-coordinate ____.
5Two circles have centres 170 units apart and radii 80 and 100. What can you say?
6Three points satisfy AB + BC = AC exactly. What does this mean?
7Why is the distance test more reliable than plotting the points and looking?
8Which of these can you now find from coordinates alone, without drawing?
More than one answer is correct.
That completes the unit on coordinates.
Noescio