11 X1 T05 02 Gradient

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Transcript of 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*