11X1 T05 02 gradient

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  • The Slope (Gradient)

  • The Slope (Gradient)vertical rise(1)

    horizontal runm

  • The Slope (Gradient)vertical rise(1)

    horizontal runm

    y

    x

  • The Slope (Gradient)vertical rise(1)

    horizontal runm

    y

    x

    l 5,5 2,1

  • The Slope (Gradient)vertical rise(1)

    horizontal runm

    y

    x

    l 5,5 2,1

    2 1

    2 1

    (2) y ymx x

  • The Slope (Gradient)vertical rise(1)

    horizontal runm

    y

    x

    l 5,5 2,1

    2 1

    2 1

    (2) y ymx x

    5 15 2l

    m

  • The Slope (Gradient)vertical rise(1)

    horizontal runm

    y

    x

    l 5,5 2,1

    2 1

    2 1

    (2) y ymx x

    5 15 2l

    m 43

  • The Slope (Gradient)vertical rise(1)

    horizontal runm

    y

    x

    l 5,5 2,1

    2 1

    2 1

    (2) y ymx x

    5 15 2l

    m 43

    L

  • The Slope (Gradient)vertical rise(1)

    horizontal runm

    y

    x

    l 5,5 2,1

    2 1

    2 1

    (2) y ymx x

    5 15 2l

    m 43

    L

    120

  • The Slope (Gradient)vertical rise(1)

    horizontal runm

    y

    x

    l 5,5 2,1

    2 1

    2 1

    (2) y ymx x

    5 15 2l

    m 43

    L

    120

    (3) tanm

  • The Slope (Gradient)vertical rise(1)

    horizontal runm

    y

    x

    l 5,5 2,1

    2 1

    2 1

    (2) y ymx x

    5 15 2l

    m 43

    L

    120

    (3) tanm is the angle of inclination

    with the positive axisx

  • The Slope (Gradient)vertical rise(1)

    horizontal runm

    y

    x

    l 5,5 2,1

    2 1

    2 1

    (2) y ymx x

    5 15 2l

    m 43

    L

    120

    (3) tanm is the angle of inclination

    with the positive axisx

    tan120Lm

  • The Slope (Gradient)vertical rise(1)

    horizontal runm

    y

    x

    l 5,5 2,1

    2 1

    2 1

    (2) y ymx x

    5 15 2l

    m 43

    L

    120

    (3) tanm is the angle of inclination

    with the positive axisx

    tan120Lm 3

  • The Slope (Gradient)vertical rise(1)

    horizontal runm

    5 15 2l

    m 43

    y

    x

    l 5,5 2,1

    2 1

    2 1

    (2) y ymx x

    L

    120

    (3) tanm is the angle of inclination

    with the positive axisx

    tan120Lm 3

  • The Slope (Gradient)vertical rise(1)

    horizontal runm

    5 15 2l

    m 43

    y

    x

    l 5,5 2,1

    2 1

    2 1

    (2) y ymx x

    L

    120

    (3) tanm is the angle of inclination

    with the positive axisx

    tan120Lm 3

    tanlm

  • The Slope (Gradient)vertical rise(1)

    horizontal runm

    5 15 2l

    m 43

    y

    x

    l 5,5 2,1

    2 1

    2 1

    (2) y ymx x

    L

    120

    (3) tanm is the angle of inclination

    with the positive axisx

    tan120Lm 3

    tanlm 4tan3

  • The Slope (Gradient)vertical rise(1)

    horizontal runm

    5 15 2l

    m 43

    y

    x

    l 5,5 2,1

    2 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

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

  • 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

  • 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

  • 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;

  • 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

  • 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

    3 12 12

    ABm

  • 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

    3 12 12

    ABm

    4 2

    32

    3

    CDm a

    a

  • 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

    3 12 12

    ABm

    4 2

    32

    3

    CDm a

    a

    If then

    AB CD

    AB CDm m

  • 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

    3 12 12

    ABm

    4 2

    32

    3

    CDm a

    a

    If then

    AB CD

    AB CDm m

    223a

  • 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

    3 12 12

    ABm

    4 2

    32

    3

    CDm a

    a

    If then

    AB CD

    AB CDm m

    223a

    3 1

    4a

    a

  • 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

    3 12 12

    ABm

    4 2

    32

    3

    CDm a

    a

    If then

    AB CD

    AB CDm m

    223a

    3 1

    4a

    a

    ( ) b AB CD

  • 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

    3 12 12

    ABm

    4 2

    32

    3

    CDm a

    a

    If then

    AB CD

    AB CDm m

    223a

    3 1

    4a

    a

    ( ) b AB CDIf then

    1AB CD

    AB CDm m

  • 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

    3 12 12

    ABm

    4 2

    32

    3

    CDm a

    a

    If then

    AB CD

    AB CDm m

    223a

    3 1

    4a

    a

    ( ) b AB CDIf then

    1AB CD

    AB CDm m

    22 13a

  • 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

    3 12 12

    ABm

    4 2

    32

    3

    CDm a

    a

    If then

    AB CD

    AB CDm m

    223a

    3 1

    4a

    a

    ( ) b AB CDIf then

    1AB CD

    AB CDm m

    22 13a

    4 3

    1a

    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*

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