Coordinates
Coordinates place points on a grid using (x, y) pairs. This topic builds on coordinate basics to cover midpoints, distance between two points, straight-line equations, and loci on grids.
๐ Learn
Midpoint of a Line Segment
Midpoint M of A(xโ, yโ) and B(xโ, yโ):
M = ((xโ+xโ)/2, (yโ+yโ)/2)
The midpoint is the average of the x-coordinates and the average of the y-coordinates.
If you know the midpoint M and one endpoint A, find the other endpoint B: B = (2Mx โ xโ, 2My โ yโ)
Length of a Line Segment
Distance d between A(xโ, yโ) and B(xโ, yโ):
d = โ[(xโโxโ)ยฒ + (yโโyโ)ยฒ]
This is Pythagoras' theorem applied to the horizontal and vertical steps between the two points.
๐ก Horizontal step = |xโโxโ|, Vertical step = |yโโyโ|. Then d = โ(hยฒ + vยฒ)
Gradient and Line Equations
Gradient: m = (yโโyโ)/(xโโxโ) = rise/run
Line through two points: y โ yโ = m(x โ xโ)
Standard form: y = mx + c (c = y-intercept)
To find the equation of a line through two points:
- Calculate gradient m
- Substitute one point into y = mx + c to find c
- Write the full equation
Parallel and Perpendicular Lines
Parallel: same gradient (mโ = mโ)
Perpendicular: gradients multiply to โ1 (mโ ร mโ = โ1), so mโ = โ1/mโ
Example: line with m = 2/3. Perpendicular gradient = โ3/2.
A horizontal line has gradient 0. A vertical line has undefined gradient.
3D Coordinates
In 3D, points are described by (x, y, z). The z-axis is perpendicular to both x and y.
3D Distance: d = โ[(xโโxโ)ยฒ + (yโโyโ)ยฒ + (zโโzโ)ยฒ]
3D Midpoint: M = ((xโ+xโ)/2, (yโ+yโ)/2, (zโ+zโ)/2)
๐ก 3D coordinates appear in cube and cuboid problems: "the vertex at (3, 0, 4)".
โ๏ธ Worked Examples
Example 1: Midpoint
Find the midpoint of A(2, 5) and B(8, โ1).
Mx = (2 + 8)/2 = 10/2 = 5
My = (5 + (โ1))/2 = 4/2 = 2
Midpoint = (5, 2)
Example 2: Distance between two points
Find the distance between P(1, 2) and Q(4, 6).
ฮx = 4 โ 1 = 3, ฮy = 6 โ 2 = 4
d = โ(3ยฒ + 4ยฒ) = โ(9 + 16) = โ25 = 5
Example 3: Equation of a line through two points
Find the equation of the line through A(1, 3) and B(4, 9).
m = (9 โ 3)/(4 โ 1) = 6/3 = 2
y = 2x + c. Substitute A(1,3): 3 = 2(1) + c โ c = 1
Equation: y = 2x + 1
Example 4: Perpendicular line
Find the equation of the line perpendicular to y = 3x โ 2 passing through (6, 1).
Gradient of given line = 3. Perpendicular gradient = โ1/3
y = โ(1/3)x + c. Substitute (6,1): 1 = โ2 + c โ c = 3
Equation: y = โ(1/3)x + 3