COORDINATE GEOMETRY

TIU Session 4

College Entrance Test Review

Rene Descartes

• Father of Modern Philosophy

• Many contributions to math: Cartesian Coordinate

System• Which uses ALGEBRA to

describe GEOMETRY Descartes’ Rule of Signs

Cartesian Coordinate System

X-axisY-axisPoint – described as an ORDERED PAIR

Recall: Undefined terms in geometry?

,y ax b

POINT – now described as (x, y)

LINE – we will define as an equation:

where x & y are unknowns, and a and b are known quantities.

PLANE – the Cartesian plane (previous slide)

“Features” of a LINE

• SLOPE

• MIDPOINT

• LENGTH OF A LINE SEGMENT (or DISTANCE BETWEEN TWO POINTS)

SLOPE OF A LINE / LINE SEGMENT

• Steepness

yslope

x

riseslope

run

1 1,x y

SLOPE – usually computed using two given points

Given the points

P(7, 2) and Q (3, -5)

in a plane. Find the slope of the line containing these two points.

•the slope m of line

is given by

=

.

2 2,x y

Graph of:Line segment with endpoints (7, 2) and (3, -5). Plotted by www.wolframalpha.com.

my

slopex

2 1

2 1

y y

x x

( 5) 2 7 7

3 7 4 4

MIDPOINT OF TWO POINTS

• Midpoint M of two points P and Q

Given the points P(7, 2) and Q(3,-5) in a

plane. Find the midpoint of

these two points.

Midpoint – is like getting the AVERAGE – is a POINT, so your answer must have two coordinates. – Get the average of the X coordinates, and the average of the Y coordinates

Midpoint Formula:

Given the points P(7, 2) and Q(3,-5) in a plane. Find the midpoint of

these two points.

1 2 1 2

Midpoint

,2 2

x x y y

7 3 2 5,

2 2

10 3

,2 2

3Midpoint 5,

2

Distance between Two Points (or the length of a line segment)

Given the points P(7, 2) and Q(3,-5) in a plane. Find the distance

between these two

points.

How do we get the length of the hypotenuse of a right triangle?

Pythagorean Theorem:

2 2 2a b c

b

a

c

Distance between two points

2 2 2c a b 2 2 2(7 3) (2 5)c 2 2 2(4) (7)c 2 16 49c 2 65c

65c

2 21 2 1 2d x x y y

Distance Formula

PRACTICE TIME!

EQUATIONS OF A LINE