Co-ordinate Geometry

Learning Outcome: •Calculate the distance between 2 points.•Calculate the midpoint of a line segment

−6 −4 −2 2 4 6 8

−2

2

4

x

y

Distance between 2 points

)( 12 yy

212

212

2 )()( yyxxd

(-1, -2)

(4, 3) ),( 22 yx

)( 12 xx

d

),( 11 yx

212

212 )()( yyxxd

Calculating the Midpoint

−6 −4 −2 2 4 6 8

−2

2

4

x

y

(-1, -2)

(4, 3) ),( 22 yx

),( 11 yx

2,

22121 yyxx

232,

241

21,

23

Co-ordinate Geometry

Learning Outcome: •Calculating the gradient of the line

joining two given points.

−8 −6 −4 −2 2 4 6 8

−4

−2

2

x

y

Gradient of a line

• Describes how steep the line is.• Given by the fraction change in y

change in x

12

12

xxyy

m

),( 22 yx

),( 11 yx(-3, -3)

(1, 2)

3132

m

45

m

Horizontal and Vertical Lines?

• The gradient of a horizontal line is zero.

• The gradient of a vertical line is undefined.

Equations of lines

• Can be written in either form:

cmxy

0 cbyax

Gradient

y - intercept

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

Rearrangement

34 xyExpress in the form ax + by + c = 0

53 xy

Express in the form y = mx + c

0632 yx

053 yx

0642 yx453

xy

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.

),( 11 yx

)( 11 xxmyy

)( 11 xxmyy

)3(22 xy622 xy82 xy 082 xyor

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).

),( 22 yx),( 11 yx

2345

m

12

12

xxyy

m

51

m

)2(514 xy

)( 11 xxmyy

2205 xy0225 xy

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 = 0

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 = 0

Proofs• When developing a coordinate geometry

proof:

1.  Draw and label the graph2.  State the formulas you will be using3.  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.

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

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

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

2 4 6 8 10

2

4

6

8

10

x

y

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.

                                  

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).

                                  

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.

                                  

                                     

Triangle ABC is shown in the diagram. Find the equation of the median through A.

Triangle ABC is shown in the diagram. Find the equation of the altitude through B.

                                     

Triangle ABC is shown in the diagram. Find the equation of the perpendicular bisector of AC.