11 x1 t05 02 gradient (2012)

42
The Slope (Gradient)

Transcript of 11 x1 t05 02 gradient (2012)

Page 1: 11 x1 t05 02 gradient (2012)

The Slope (Gradient)

Page 2: 11 x1 t05 02 gradient (2012)

The Slope (Gradient)vertical rise(1)

horizontal runm

Page 3: 11 x1 t05 02 gradient (2012)

The Slope (Gradient)vertical rise(1)

horizontal runm

y

x

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The Slope (Gradient)vertical rise(1)

horizontal runm

y

x

l 5,5

2,1

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The Slope (Gradient)vertical rise(1)

horizontal runm

y

x

l 5,5

2,12 1

2 1

(2) y ymx x

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The Slope (Gradient)vertical rise(1)

horizontal runm

y

x

l 5,5

2,12 1

2 1

(2) y ymx x

5 15 2lm

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The Slope (Gradient)vertical rise(1)

horizontal runm

y

x

l 5,5

2,12 1

2 1

(2) y ymx x

5 15 2lm

43

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The Slope (Gradient)vertical rise(1)

horizontal runm

y

x

l 5,5

2,12 1

2 1

(2) y ymx x

5 15 2lm

43

L

Page 9: 11 x1 t05 02 gradient (2012)

The Slope (Gradient)vertical rise(1)

horizontal runm

y

x

l 5,5

2,12 1

2 1

(2) y ymx x

5 15 2lm

43

L

120

Page 10: 11 x1 t05 02 gradient (2012)

The Slope (Gradient)vertical rise(1)

horizontal runm

y

x

l 5,5

2,12 1

2 1

(2) y ymx x

5 15 2lm

43

L

120

(3) tanm

Page 11: 11 x1 t05 02 gradient (2012)

The Slope (Gradient)vertical rise(1)

horizontal runm

y

x

l 5,5

2,12 1

2 1

(2) y ymx x

5 15 2lm

43

L

120

(3) tanm is the angle of inclination

with the positive axisx

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The Slope (Gradient)vertical rise(1)

horizontal runm

y

x

l 5,5

2,12 1

2 1

(2) y ymx x

5 15 2lm

43

L

120

(3) tanm is the angle of inclination

with the positive axisx

tan120Lm

Page 13: 11 x1 t05 02 gradient (2012)

The Slope (Gradient)vertical rise(1)

horizontal runm

y

x

l 5,5

2,12 1

2 1

(2) y ymx x

5 15 2lm

43

L

120

(3) tanm is the angle of inclination

with the positive axisx

tan120Lm

3

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The Slope (Gradient)vertical rise(1)

horizontal runm

5 15 2lm

43

y

x

l 5,5

2,12 1

2 1

(2) y ymx x

L

120

(3) tanm is the angle of inclination

with the positive axisx

tan120Lm

3

Page 15: 11 x1 t05 02 gradient (2012)

The Slope (Gradient)vertical rise(1)

horizontal runm

5 15 2lm

43

y

x

l 5,5

2,12 1

2 1

(2) y ymx x

L

120

(3) tanm is the angle of inclination

with the positive axisx

tan120Lm

3

tanlm

Page 16: 11 x1 t05 02 gradient (2012)

The Slope (Gradient)vertical rise(1)

horizontal runm

5 15 2lm

43

y

x

l 5,5

2,12 1

2 1

(2) y ymx x

L

120

(3) tanm is the angle of inclination

with the positive axisx

tan120Lm

3

tanlm 4tan3

Page 17: 11 x1 t05 02 gradient (2012)

The Slope (Gradient)vertical rise(1)

horizontal runm

5 15 2lm

43

y

x

l 5,5

2,12 1

2 1

(2) y ymx x

L

120

(3) tanm is the angle of inclination

with the positive axisx

tan120Lm

3

tanlm 4tan3

53

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Two lines are parallel iff they have the same slope1 2i.e. m m

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Two lines are parallel iff they have the same slope1 2i.e. m m

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

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Two lines are parallel iff they have the same slope1 2i.e. m m

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

1 2i.e. 1m m

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Two lines are parallel iff they have the same slope1 2i.e. m m

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

1 2i.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;

Page 22: 11 x1 t05 02 gradient (2012)

Two lines are parallel iff they have the same slope1 2i.e. m m

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

1 2i.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

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Two lines are parallel iff they have the same slope1 2i.e. m m

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

1 2i.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

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Two lines are parallel iff they have the same slope1 2i.e. m m

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

1 2i.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 23

23

CDma

a

Page 25: 11 x1 t05 02 gradient (2012)

Two lines are parallel iff they have the same slope1 2i.e. m m

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

1 2i.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 23

23

CDma

a

If then

AB CD

AB CDm m

Page 26: 11 x1 t05 02 gradient (2012)

Two lines are parallel iff they have the same slope1 2i.e. m m

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

1 2i.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 23

23

CDma

a

If then

AB CD

AB CDm m

223a

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Two lines are parallel iff they have the same slope1 2i.e. m m

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

1 2i.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 23

23

CDma

a

If then

AB CD

AB CDm m

223a

3 14

aa

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Two lines are parallel iff they have the same slope1 2i.e. m m

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

1 2i.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 23

23

CDma

a

If then

AB CD

AB CDm m

223a

3 14

aa

( ) b AB CD

Page 29: 11 x1 t05 02 gradient (2012)

Two lines are parallel iff they have the same slope1 2i.e. m m

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

1 2i.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 23

23

CDma

a

If then

AB CD

AB CDm m

223a

3 14

aa

( ) b AB CD

If then1AB CD

AB CDm m

Page 30: 11 x1 t05 02 gradient (2012)

Two lines are parallel iff they have the same slope1 2i.e. m m

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

1 2i.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 23

23

CDma

a

If then

AB CD

AB CDm m

223a

3 14

aa

( ) b AB CD

If then1AB CD

AB CDm m

22 13a

Page 31: 11 x1 t05 02 gradient (2012)

Two lines are parallel iff they have the same slope1 2i.e. m m

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

1 2i.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 23

23

CDma

a

If then

AB CD

AB CDm m

223a

3 14

aa

( ) b AB CD

If then1AB CD

AB CDm m

22 13a

4 31a

a

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(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.

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(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

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(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

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(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

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(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

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(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

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

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

( ) find ABi m

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

( ) find ABi m

( ) find ACii m

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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

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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*