11 x1 t05 02 gradient

The Slope (Gradient)

The Slope (Gradient)vertical rise

(1) horizontal run

m

The Slope (Gradient)vertical rise

(1) horizontal run

my

x

The Slope (Gradient)vertical rise

(1) horizontal run

my

x

l 5,5

2,1

The Slope (Gradient)vertical rise

(1) horizontal run

my

x

l 5,5

2,12 1

2 1

(2) y y

mx x

The Slope (Gradient)vertical rise

(1) horizontal run

my

x

l 5,5

2,12 1

2 1

(2) y y

mx x

5 15 2lm

The Slope (Gradient)vertical rise

(1) horizontal run

my

x

l 5,5

2,12 1

2 1

(2) y y

mx x

5 15 2lm

43

The Slope (Gradient)vertical rise

(1) horizontal run

my

x

l 5,5

2,12 1

2 1

(2) y y

mx x

5 15 2lm

43

L

The Slope (Gradient)vertical rise

(1) horizontal run

my

x

l 5,5

2,12 1

2 1

(2) y y

mx x

5 15 2lm

43

L

120

The Slope (Gradient)vertical rise

(1) horizontal run

my

x

l 5,5

2,12 1

2 1

(2) y y

mx x

5 15 2lm

43

L

120

(3) tanm

The Slope (Gradient)vertical rise

(1) horizontal run

my

x

l 5,5

2,12 1

2 1

(2) y y

mx x

5 15 2lm

43

L

120

(3) tanm is the angle of inclination

with the positive axisx

The Slope (Gradient)vertical rise

(1) horizontal run

my

x

l 5,5

2,12 1

2 1

(2) y y

mx x

5 15 2lm

43

L

120

(3) tanm is the angle of inclination

with the positive axisx

tan120Lm

The Slope (Gradient)vertical rise

(1) horizontal run

my

x

l 5,5

2,12 1

2 1

(2) y y

mx x

5 15 2lm

43

L

120

(3) tanm is the angle of inclination

with the positive axisx

tan120Lm

3

The Slope (Gradient)vertical rise

(1) horizontal run

m

5 15 2lm

43

y

x

l 5,5

2,12 1

2 1

(2) y y

mx x

L

120

(3) tanm is the angle of inclination

with the positive axisx

tan120Lm

3

The Slope (Gradient)vertical rise

(1) horizontal run

m

5 15 2lm

43

y

x

l 5,5

2,12 1

2 1

(2) y y

mx x

L

120

(3) tanm is the angle of inclination

with the positive axisx

tan120Lm

3

tanlm

The Slope (Gradient)vertical rise

(1) horizontal run

m

5 15 2lm

43

y

x

l 5,5

2,12 1

2 1

(2) y y

mx x

L

120

(3) tanm is the angle of inclination

with the positive axisx

tan120Lm

3

tanlm 4

tan3

The Slope (Gradient)vertical rise

(1) horizontal run

m

5 15 2lm

43

y

x

l 5,5

2,12 1

2 1

(2) y y

mx x

L

120

(3) tanm is the angle of inclination

with the positive axisx

tan120Lm

3

tanlm 4

tan3

53

Two lines are parallel iff they have the same slope

1 2 i.e. m m

Two lines are parallel iff they have the same slope

1 2 i.e. m m

Two lines are perpendicular iff their slopes are thenegative inverse of each other

Two lines are parallel iff they have the same slope

1 2 i.e. m m

Two lines are perpendicular iff their slopes are thenegative inverse of each other

1 2 i.e. 1m m

Two lines are parallel iff they have the same slope

1 2 i.e. m m

Two lines are perpendicular iff their slopes are thenegative inverse of each other

1 2 i.e. 1m m

e.g. A, B, C and D are the points (1,1), (2,3), (3,2) and (a,4) Find a such that;

Two lines are parallel iff they have the same slope

1 2 i.e. m m

Two lines are perpendicular iff their slopes are thenegative inverse of each other

1 2 i.e. 1m m

e.g. A, B, C and D are the points (1,1), (2,3), (3,2) and (a,4) Find a such that;

( ) a AB CD

Two lines are parallel iff they have the same slope

1 2 i.e. m m

Two lines are perpendicular iff their slopes are thenegative inverse of each other

1 2 i.e. 1m m

e.g. A, B, C and D are the points (1,1), (2,3), (3,2) and (a,4) Find a such that;

( ) a AB CD3 12 12

ABm

Two lines are parallel iff they have the same slope

1 2 i.e. m m

Two lines are perpendicular iff their slopes are thenegative inverse of each other

1 2 i.e. 1m m

e.g. A, B, C and D are the points (1,1), (2,3), (3,2) and (a,4) Find a such that;

( ) a AB CD3 12 12

ABm

4 2

32

3

CDm a

a

Two lines are parallel iff they have the same slope

1 2 i.e. m m

Two lines are perpendicular iff their slopes are thenegative inverse of each other

1 2 i.e. 1m m

e.g. A, B, C and D are the points (1,1), (2,3), (3,2) and (a,4) Find a such that;

( ) a AB CD3 12 12

ABm

4 2

32

3

CDm a

a

If then

AB CD

AB CD

m m

Two lines are parallel iff they have the same slope

1 2 i.e. m m

Two lines are perpendicular iff their slopes are thenegative inverse of each other

1 2 i.e. 1m m

e.g. A, B, C and D are the points (1,1), (2,3), (3,2) and (a,4) Find a such that;

( ) a AB CD3 12 12

ABm

4 2

32

3

CDm a

a

If then

AB CD

AB CD

m m

22

3a

Two lines are parallel iff they have the same slope

1 2 i.e. m m

Two lines are perpendicular iff their slopes are thenegative inverse of each other

1 2 i.e. 1m m

e.g. A, B, C and D are the points (1,1), (2,3), (3,2) and (a,4) Find a such that;

( ) a AB CD3 12 12

ABm

4 2

32

3

CDm a

a

If then

AB CD

AB CD

m m

22

3a

3 1

4

a

a

Two lines are parallel iff they have the same slope

1 2 i.e. m m

Two lines are perpendicular iff their slopes are thenegative inverse of each other

1 2 i.e. 1m m

e.g. A, B, C and D are the points (1,1), (2,3), (3,2) and (a,4) Find a such that;

( ) a AB CD3 12 12

ABm

4 2

32

3

CDm a

a

If then

AB CD

AB CD

m m

22

3a

3 1

4

a

a

( ) b AB CD

Two lines are parallel iff they have the same slope

1 2 i.e. m m

Two lines are perpendicular iff their slopes are thenegative inverse of each other

1 2 i.e. 1m m

e.g. A, B, C and D are the points (1,1), (2,3), (3,2) and (a,4) Find a such that;

( ) a AB CD3 12 12

ABm

4 2

32

3

CDm a

a

If then

AB CD

AB CD

m m

22

3a

3 1

4

a

a

( ) b AB CD

If then

1AB CD

AB CD

m m

Two lines are parallel iff they have the same slope

1 2 i.e. m m

Two lines are perpendicular iff their slopes are thenegative inverse of each other

1 2 i.e. 1m m

e.g. A, B, C and D are the points (1,1), (2,3), (3,2) and (a,4) Find a such that;

( ) a AB CD3 12 12

ABm

4 2

32

3

CDm a

a

If then

AB CD

AB CD

m m

22

3a

3 1

4

a

a

( ) b AB CD

If then

1AB CD

AB CD

m m

22 1

3a

Two lines are parallel iff they have the same slope

1 2 i.e. m m

Two lines are perpendicular iff their slopes are thenegative inverse of each other

1 2 i.e. 1m m

e.g. A, B, C and D are the points (1,1), (2,3), (3,2) and (a,4) Find a such that;

( ) a AB CD3 12 12

ABm

4 2

32

3

CDm a

a

If then

AB CD

AB CD

m m

22

3a

3 1

4

a

a

( ) b AB CD

If then

1AB CD

AB CD

m m

22 1

3a

4 3

1

a

a

(ii) A(4,0), B(1,5) and C(–3,–5) are three vertices of a parallelogram ABCD. Find the coordinates of D, the fourth vertex of the parallelogram.

(ii) A(4,0), B(1,5) and C(–3,–5) are three vertices of a parallelogram ABCD. Find the coordinates of D, the fourth vertex of the parallelogram. y

x 4,0A

3, 5C

1,5B

(ii) A(4,0), B(1,5) and C(–3,–5) are three vertices of a parallelogram ABCD. Find the coordinates of D, the fourth vertex of the parallelogram. y

x 4,0A

3, 5C

1,5B

,D x y

3

5

(ii) A(4,0), B(1,5) and C(–3,–5) are three vertices of a parallelogram ABCD. Find the coordinates of D, the fourth vertex of the parallelogram. y

x 4,0A

3, 5C

1,5B

,D x y

3

5

3

5

(ii) A(4,0), B(1,5) and C(–3,–5) are three vertices of a parallelogram ABCD. Find the coordinates of D, the fourth vertex of the parallelogram. y

x 4,0A

3, 5C

1,5B

,D x y

3

5

3

5

3 3, 5 5D

(ii) A(4,0), B(1,5) and C(–3,–5) are three vertices of a parallelogram ABCD. Find the coordinates of D, the fourth vertex of the parallelogram. y

x 4,0A

1,5B

3, 5C

,D x y

3

5

3

5

3 3, 5 5D

0, 10

To prove three points (A, B, C) are collinear;

To prove three points (A, B, C) are collinear;

( ) find ABi m

To prove three points (A, B, C) are collinear;

( ) find ABi m

( ) find ACii m

To prove three points (A, B, C) are collinear;

( ) find ABi m

( ) find ACii m

( ) if then , , are collinearAB ACiii m m A B C

To prove three points (A, B, C) are collinear;

( ) find ABi m

( ) find ACii m

( ) if then , , are collinearAB ACiii m m A B C

Exercise 5B; 2ace, 3bd, 4 ii, iv, 5ae, 8, 9a, 11ac, 13, 15, 18, 19, 20, 24*