Lecture 1 (Part A) Factorization resources/Mathematics Conten… · Straight Line Coordinate...

NCS Mathematics

DVD Series

Straight Line

Coordinate

Geometry

Outcomes for this DVD

Recall and apply distance, gradient and midpoint formulae.

LE SSON

Derive and apply a formu

1

In this DVD we will :

la for the inclination of a line.

Derive and apply formulae for the equation of a line.

LESSO

2

N

L E SS ON 3

NCS Mathematics

DVD Series

Lesson 1

Distance,

Gradient and

Midpoint

Formulae

Distance, Gradient and Midpoint Formulae

Gradient of line :AB

Co-ordinates of midpoint of segment :AB

Distance between and :A B

1 2 1 2;2 2

x x y y

1 1 2 2You should know that for two points ; and ; :A x y B x y

1 1;A x y

2 2;x y B

2 2

2 1 2 1AB x x y y

2 1

2 1

tanAB

y yym

x x x

y

x

Application of Distance Formula

2;6 , 5;2 , 4; 4 and 8;2

are the vertices of quadrilateral .

A B C D

ABCD

A

B

C

D

2 2

5 2 6 2AB

49 16 65

2 24 6 16 36 52DC

is not a parallelogram nor a rectangle.ABCD

Application of Gradient Formula

2;6 , 5;2 , 4; 4 and 8;2


A B C D

ABCD

A

C

D2 6 4 4

5 2 7 7ABm

Angle between and the

positive axis is acute.

AB

x

4 2

6 3ADm

6 2and

9 3BCm

//AD BC

6 3

4 2CDm

and CD AD CD BC

B

is a

Trapezium

ABCD

1 and 1AD CD BC CD

m m m m

Know: AB CD

Application of Midpoint Formula

2;6 , 5;2 , 4; 4 and 8;2


A B C D

ABCD

A

B

C

D

2 4 6 4

Midpoint of is ; 3;12 2

AC

5 8 2 2 3Midpoint of is ; ;2

2 2 2BD

Diagonals are not bisecting

each other.

Tutorial 1: Distance, Gradient

and Midpoint Formulae Given a quadrilateral with vertices as indicated.

2;4A

1;2B

5;4C

6;6D

1) Determine whether the opposite

sides are equal in length.


sides are parallel.

3) Determine whether the diagonals

bisect each other.

PAUSE DVD

• Do Tutorial 1

• Then View Solutions

4) Classify this quadrilateral.

5) Can this quadrilateral be

classified as a rectangle?

Tutorial 1 Problem 1: Suggested Solution

2;4A

1;2B

5;4C

6;6D


sides are equal in length.

1 4 5AB

and 1 4 5CD

AB CD

Similarily 16 4 20AD BC

Opposite sides are equal.


2;4A

1;2B

5;4C

6;6D


sides are parallel.

4 22

2 1ABm

6 4

and 26 5CD

m

// AB CD

mB C mA D

2 1

4 2AD BCm m

Gradients

are eq/

u/

l

a AD BC


2;4A

1;2B

5;4C

6;6D

3) Determine whether

the diagonals bisect

each other.

Midpoint of diagonal

2 5 4 4 7is ; ;4

2 2 2

AC

Midpoint of diagonal

1 6 2 6 7is ; ;4

2 2 2

BD

Diagonals bisect each other

7 at the point ;4

2


2;4A

1;2B

5;4C

6;6D

4) Classify this quadrilateral.

We showed in question 2 that:

// and //AB DC AD BC

can be classified as a parallelogram.ABCD


5) Can this quadrilateral be

classified as a rectangle?

We showed in question 2 that:

12 and

2AB BCm m

1AB BC

m m

90B is not a rectangle.ABCD

In question 1 we showed that

opposite sides are equal.

NCS Mathematics

DVD Series

Lesson 2

Inclination

of a Line

What is the Inclination of a Line?

The angle that a line makes with the positive direction

of the axis is called th inclinatione of the line.x

x

y

y

x

A

Btan

AB

ym

x

for 0 180

is the Inclination of the line

When are two Lines Parallel?

Two lines with the same

inclination will be parallel.

x

y

A

B

C

D

// tanAB CD

AB CD m m

When are two Lines Perpendicular?

x

y

A

B

C

D

tan and tanAB CDm m

90

tan tanAB CDm m tan tan 90

tan cot 1

1AB CD

AB CD m m

Assume that

AB CD

Transforming Inclination into Gradient

Given that the inclination of a line is 78 ,

find the gradient of the line.

Example :

Line tan 78 4,705m

Given that the inclination of a line is 158 ,

find the gradient of the line.

Example :

Line tan158 tan 180 22

tan 22 0,404

m

Transforming Gradient to Inclination

Find the inclination of a line if the

3 the gradient of the line is .

4

Example :

3tan tan

4m 1 3

tan 36,874

Find the inclination of a line if the

5 the gradient of the line is .

6

Example :

5tan

6 1 5

tan 39,816

Reference Angle

180 39,81 140,19

Tutorial 2: Inclination of a Line

1) Given that // and that 1,7.

Determine the inclination of line .

ABAB CD m

CD

2) The inclination of line is 73 .

Determine the inclination of line

if .

AB

CD CD AB

PAUSE DVD

• Do Tutorial 2



1) Given that // and that 1,7.

Determine the inclination of line .

ABAB CD m

CD

1,7CD

m

tan 1,7

1tan 1,7 59,5

Inclination of is 59,5CD



Determine the inclination of line if

.

AB

CD

CD AB

and tan 73AB

m 1

CD

AB

mm

1

tan 73CDm

1 1 Inclination of is 180 tan

tan 73CD

or 180 17 or 163

Tutorial 2 Problem 2: Alternative Solution


Determine the inclination of line if

.

AB

CD

CD AB

x

y

A

B

73

C

D

90 73 163

NCS Mathematics

DVD Series

Lesson 3

Finding the

Equation of a

Straight Line

Point – Gradient Form of a Straight Line

1 1Let ; be a fixed point on a line with gradient .A x y m

x

y

1 1;A x y

m

Let ; be any point on the line.B x y

;B x y

1;C x y

y CBm

x AC

1

1

y y

x x

1 1y y m x x

Point Gradient form of straight line

Utilize the Point-Gradient Formula

Find the equation of the line passing through

2the point 3; 5 that has a gradient of .

3

Use the point-gradient formula.

1 1y y m x x

2

Equation of line is given by 5 33

y x

2 2

or 2 5 or 33 3

x xy y

Standard form of a straight line.

2 – Point Form of a Straight line

1 1 2 2Let ; and ; be two fixed points on a line.A x y B x y

2 1

2 1

is the gradient of this line.y yy

mx x x

1 1

2 11 1

2 1

y y m x x

y yy y x x

x x

Two Point form of straight line

Use point gradient form of line:

Utilize the two - point formula

Find the equation of the line passing through

the points 3; 5 and 4;3 .

Use the two point formula:

2 11 1

2 1

y yy y x x

x x

5 3

Equation of line is given by 3 43 4

y x

8 8 11

or 3 4 or 7 7 7

xy x y

Alternatively:

85 3

7y x

Tutorial 3: Part 1: Equations of a Straight Line

1) Find the equation of the line that:

(a) Is parallel to the line 3 2

and passes through 3; 4 .

(b) Is perpendicular to the line

2 3 0 and passes through

3

y x

xy

the point 4;5 .

PAUSE DVD

• Do Tutorial 3 (Part 1)


Tutorial 3 Problem 1(a): Suggested Solution

(1a) Find the equation of the line that is parallel to

the line 3 2 and passes through 3; 4 .y x

3 is the gradient of the line.m

Use the point gradient formula:

1 1y y m x x

Equation of line is given by 4 3 3y x

or 3 5y x

Tutorial 3 Problem 1(b): Suggested Solution

(1b) Find the equation of the line that is perpendicular to the

2 line 3 0 and passes through the point 4;5 .

3

xy

2 23 Gradient of given line is

3 3

xy

3 Gradient of line perpendicular to given line is

2

From point gradient formula the equation of required

line is given by:

3

5 42

y x 3

or 112

xy

Tutorial 3: Part 2: Equations of a Straight Line

2) 2; 4 , 4;2 and 7;1

are the vertices of and

is the midpoint of .

(a) Find the equation of line .

(b) Is ?

(c) Determine the equation of the

P Q R

PQR

M PQ

MR

MR PQ

line through that is perpendicular to .

(d) Find the equation of the line that is parallel

to and passes through .

M PQ

PR M

PAUSE DVD

• Do Tutorial 3 (Part 2)


Tutorial 3 Problems 2 (a) and (b): Suggested Solutions

2) 2; 4 , 4;2 and 7;1 are the vertices of and


(a) Find the equation of line .

(b) Is ?

P Q R PQR

M PQ

MR

MR PQ

2; 4P

4;2Q

7;1R 3; 1M 3; 1M

2 1

10 5MRm

1

Equation of : 1 75

MR y x 2

or 5 5

xy

63

2PQm

MR PQ

2 a

2 b Reason?

Tutorial 3 Problem 2 (c): Suggested Solutions

2) 2; 4 , 4;2 and 7;1



(c) Determine the equation of the line

through that is perpendicular to .

P Q R

PQR

M PQ

M PQ

2 c 3PQ

m 1 Gradient of line to

3PQ

Equation of requested line:

Know: 3; 1M 1

1 33

y x or 3

xy

Tutorial 3 Problem 2 (d): Suggested Solutions

2) 2; 4 , 4;2 and 7;1



(d) Find the equation of the line that

is parallel to and passes through .

P Q R

PQR

M PQ

PR M

Know: 3; 1M

2 dLINE

5 5

9 9PRm m

Equation: 5 2

or 13

3 9

5

9

xyy x

End of the DVD on

Straight Line Coordinate Geometry

REMEMBER!

•Consult text-books for additional examples.

•Attempt as many as possible other similar examples

on your own.

•Compare your methods with those that were

discussed in the DVD.

•Repeat this procedure until you are confident.

•Do not forget:

Practice makes perfect!

Lecture 1 (Part A) Factorization resources/Mathematics Conten… · Straight Line Coordinate...

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Transcript of Lecture 1 (Part A) Factorization resources/Mathematics Conten… · Straight Line Coordinate...