Co-ordinate Geometry - PowerPoint Presentation

Slide1

Presentation of Coordinate Geometry

Design

by Ms

Sheema

AftabSlide2

Co-ordinate Geometry

Learning Outcome:

Calculate the distance between 2 points.

Calculate the midpoint of a line segmentSlide3

Distance between 2 points

(-1, -2)

(4, 3)

dSlide4

Calculating the Midpoint

(-1, -2)

(4, 3)Slide5

Co-ordinate Geometry

Learning Outcome:

Calculating the gradient of the line joining two given points.Slide6

Gradient of a line

Describes how steep the line is.

Given by the fraction

change in y

change in x

(-3, -3)

(1, 2)Slide7

Horizontal and Vertical Lines?

The gradient of a horizontal line is zero.

The gradient of a vertical line is undefined.Slide8

Equations of lines

Can be written in either form:

Gradient

y - intercept

The x term is to be written first, with a positive coefficient.Slide9

Rearrangement

Express in the form ax + by + c = 0

Express in the form y = mx + cSlide10

Given gradient m and a point

The equation of the line is

This is called the point-gradient formula.

Find the equation of the line that passes through (3,-2) with the gradient of 2.

orSlide11

Given two points

Find the equation of this line.

First find the gradient, then use the point gradient formula.

Find the equation of the line joining the points (-2, 4 ) and (3, 5).Slide12

Parallel Lines

Have the same gradient

Will never meet

Find the equation of the line that passes through the point (3, -13) that is parallel to the line y + 3x – 2 = 0Slide13

Perpendicular Lines

Two lines are perpendicular if they meet at right-angles

Gradients multiply together to equal -1 (except if you have a horizontal line).

Each gradient is the negative reciprocal of the other.

Find the equation of the line that passes through the point (6, -5) that is perpendicular to the line 2x – 3y – 5 = 0Slide14

Proofs

When developing a coordinate geometry proof:

1.  Draw and label the graph

2.  State the formulas you will be using

3.  Show ALL work (if you are using your graphing calculator, be sure to show your screen displays as part of your work.)

4.  Have a concluding sentence stating what you have proven and why it is true.Slide15

Collinear points

Points are collinear if they all lie on the same line.

You need to establish that they have

a common direction (equal gradients)

a common point

Prove that P(1,4), Q(4, 6) and R(10, 10) are collinear Slide16

The line segments have a common direction (gradients =2/3)

and a common point (P) so P, Q and R are collinear. Slide17

Median

A

median

is the line that joins a

vertex

of a triangle to the midpoint

of the opposite side.

The diagram shows all three medians which are concurrent at a point called the centroid.

                                 Slide18

Perpendicular Bisector

A

perpendicular bisector

is the line that passes through the

midpoint

of a side and is

perpendicular (at right-angles) to that side.

The diagram shows all three perpendicular bisectors which are concurrent at a point called the circumcentre (the centre of the surrounding circle).

                                 Slide19

Altitude

An

altitude

is the line that joins a

vertex

of a triangle to the opposite side, and is

perpendicular to that side.

The diagram shows all three altitudes which are concurrent at a point called the orthocentre.

                                 Slide20

Triangle ABC is shown in the diagram.

Find the equation of the

median

through A.Slide21

Triangle ABC is shown in the diagram.

Find the equation of the

altitude

through B.

                                     Slide22

Triangle ABC is shown in the diagram.

Find the equation of the

perpendicular bisector

of AC.