11 x1 t05 02 gradient (2013)
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The Slope (Gradient)
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Transcript of 11 x1 t05 02 gradient (2013)
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
2,12 1
2 1
(2) y ymx x
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
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
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
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
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
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
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
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
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
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
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
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
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 CD3 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 CD3 12 12
ABm
4 23
23
CDma
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 CD3 12 12
ABm
4 23
23
CDma
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 CD3 12 12
ABm
4 23
23
CDma
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 CD3 12 12
ABm
4 23
23
CDma
a
If then
AB CD
AB CDm m
223a
3 14
aa
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
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
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
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
(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*
