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THE LINE

Exam Cog
①THE CARTESIAN PLANE

Point = (x, y)   Origin = (0,0)

x first: left −, right +   y second: down −, up +

★ Along the corridor, then up the stairs (x then y)
②MIDPOINT

M = ( (x₁+x₂)/2 , (y₁+y₂)/2 )

The average of the two ends.

Backwards: know M + one end → repeat the jump for the other end.

③DISTANCE |AB|

|AB| = √[(x₂−x₁)² + (y₂−y₁)²]

It is Pythagoras on the run & rise.

⚠ WATCH: subtract in the SAME order top and bottom; square kills the sign.
④SLOPE m

m = rise/run = (y₂−y₁)/(x₂−x₁)

Up-right → +   Down-right → −   Flat → 0   Vertical → none

★ Same slope = parallel lines
⑤EQUATION OF A LINE

Two forms:

y = mx + c  (c = y-intercept)

y − y₁ = m(x − x₁)

From 2 points: find m first, then use point−slope.

⑥PARALLEL & PERPENDICULAR

Parallel: m₁ = m₂

Perpendicular: m₁ · m₂ = −1

✓ TIP: perpendicular slope = flip & change sign (negative reciprocal).
⑦INTERCEPTS

x-intercept: let y = 0, solve for x

y-intercept: let x = 0, solve for y

⑧IS THE POINT ON THE LINE?

Substitute the point into the equation.

Both sides equal → the point is ON the line.

⑨EXAM METHOD
  1. Read it — draw & label the points
  2. Name x₁,y₁,x₂,y₂
  3. Pick the formula (all three are in your tables)
  4. State units where lengths are asked
★ A quick sketch stops sign mistakes
⑩KEY VOCAB

Slope = steepness  Intercept = where it cuts an axis

Collinear = points on one straight line (equal slopes)