high school Plane Coordinates & Line Equations

Plane Coordinates & Line Equations

On the coordinate plane, two points already fix their distance: the horizontal and vertical gaps make a right triangle whose hypotenuse is that distance, by Pythagoras. The midpoint is just the average of the two points' coordinates on each axis. A line, in turn, is set entirely by its slope and y-intercept. Drag B and the distance and its triangle change; adjust slope and intercept and the line pivots and slides.

Distance & Midpoint of Two Points

On the coordinate plane the distance uses Pythagoras, the midpoint is the average of coordinates.

B's x4

B's y3

💡 Intuition for Distance & Midpoint

①Distance = hypotenuse of a right triangle (Pythagoras)

②Midpoint = average of the two coordinates

③Drawing the Δx, Δy right triangle makes it clear

④In this sketch A is fixed at (1, 1) and only B moves

Distance & Midpoint Formulas

Distance Between Two Points

d = √((x₂ - x₁)² + (y₂ - y₁)²)

Coordinate version of Pythagoras

Midpoint

M = (x₁ + x₂2, y₁ + y₂2)

Arithmetic mean of each coordinate

Line Equation — Slope & y-intercept

Slope m2

y-intercept b-1

Slope-Intercept Form

y = mx + b

m: slope (Δy/Δx), b: y-intercept

Point + Slope

y - y₁ = m(x - x₁)

Line through (x₁, y₁) with slope m

Other Line Forms & Point-Line Distance

General Form

ax + by + c = 0

Covers every line (including vertical)

Distance from Point to Line

d = |ax₁ + by₁ + c|√(a² + b²)

From (x₁, y₁) to ax + by + c = 0

📐 Two Lines

①Parallel: m₁ = m₂

②Perpendicular: m₁ · m₂ = -1

③Identical: same slope and intercept

Work It Out

Example 1

Find the point P that divides A(1, 2) and B(7, 8) internally in the ratio 2 : 1.

Apply the internal-division formula with m : n = 2 : 1.

P = ( (2·7 + 1·1)/3, (2·8 + 1·2)/3 )

Compute.

P = (15/3, 18/3) = (5, 6)

▸ P(5, 6)

In ratio m : n, multiply the far endpoint by m and the near one by n.

Example 2

For A(−1, 3) and B(3, −1), find the length of segment AB and its midpoint M.

Use the distance and midpoint formulas.

AB = √((3 − (−1))² + (−1 − 3)²) = √(16 + 16)

Simplify.

AB = √32 = 4√2, M = ((−1 + 3)/2, (3 + (−1))/2) = (1, 1)

▸ AB = 4√2, midpoint M(1, 1)

For distance and midpoint, just substitute the coordinates.

Wrap-up

Distance Formula

d = √((x₂ - x₁)² + (y₂ - y₁)²)

Core tool on the coordinate plane

Point-to-Line Distance

d = |ax₀ + by₀ + c|√(a² + b²)

Shortest distance to the foot of perpendicular

Grade-10 school exam type

Find the distance from the point (2, 3) to the line 3x − 4y + 1 = 0.

①1

②2

③3

④4

⑤5

▸ ① 1

Use the point-to-line distance formula.

d = |a·x₁ + b·y₁ + c| / √(a² + b²)

Substitute the values.

d = |3·2 − 4·3 + 1| / √(9 + 16) = |−5| / 5 = 1

🎯 Exam Points

①Distance: square root of sum of squared differences

②Midpoint: special case of internal division (1:1)

③Slope = tanθ — angle of inclination

④Point-line distance: absolute value top, square root bottom

⑤m₁m₂ = -1 (perpendicularity) is asked frequently

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Various Inequalities

Equations of a Circle

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