Coordinate Geometry

Basic Formulas: 

To find a side: AB^2=BC^2+AC^2 ....... AB = sqr(( y2-y1)^2 + (x2-x1)^2)

Gradient: y2-y1/x2-x1 

Perpendicular lines: m1xm2=-1 

Midpoint: (x1+x2/2) (y1+y2/2) 

Show that A(3,4), B(2,7) and c(8,9) are coordinates of a rectangle.

And find point D.

1. Prove that BC and AB are at 90degrees. 

BC = 9-7/8-2 = 1/3      &     AB = 7-4/2-3 = -3 

ANGLE = 1/3 x -3 = -1!! so they are at 90degrees, hence right angles.

2. Find vector BC : (6/2) 

so because BC and AD are paralell, AD would have the same vector: (6/2) 

to find D.... A(3,4)....3+6, 4+2....D=(9,6) 

Method 1.

y=mx+c 

Find equation of the line through (2,-3) with gradient 3/4 in form ax+by+c=0

Sub in gradient.....y=3/4x+c

Find x and y.....x=2 y=-3 

Sub x and y into equation....-3=3/4x2+c

Find c.....c=-4,1/2

Put into format...y=3/4x-4,1/2....(x4)

4y=3x-18 

0=3x-4y-18

Method 2 

(y-y1) = m(x-x1)

Find equation of the line through (2,-3) with gradient 3/4 in form ax+by+c=0

(y--3) = 3/4 (x-2) 

y+3 =3/4x-6/4 

y+3=3/4x-6/4-12/4

4y=3x-6-12

0=3x-4y-18