11 x1 t05 02 gradient (2012)
TRANSCRIPT
![Page 1: 11 x1 t05 02 gradient (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/1.jpg)
The Slope (Gradient)
![Page 2: 11 x1 t05 02 gradient (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/2.jpg)
The Slope (Gradient)vertical rise(1)
horizontal runm
![Page 3: 11 x1 t05 02 gradient (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/3.jpg)
The Slope (Gradient)vertical rise(1)
horizontal runm
y
x
![Page 4: 11 x1 t05 02 gradient (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/4.jpg)
The Slope (Gradient)vertical rise(1)
horizontal runm
y
x
l 5,5
2,1
![Page 5: 11 x1 t05 02 gradient (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/5.jpg)
The Slope (Gradient)vertical rise(1)
horizontal runm
y
x
l 5,5
2,12 1
2 1
(2) y ymx x
![Page 6: 11 x1 t05 02 gradient (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/6.jpg)
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
![Page 7: 11 x1 t05 02 gradient (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/7.jpg)
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
![Page 8: 11 x1 t05 02 gradient (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/8.jpg)
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)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/9.jpg)
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)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/10.jpg)
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)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/11.jpg)
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
![Page 12: 11 x1 t05 02 gradient (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/12.jpg)
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)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/13.jpg)
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
![Page 14: 11 x1 t05 02 gradient (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/14.jpg)
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)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/15.jpg)
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)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/16.jpg)
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)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/17.jpg)
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
![Page 18: 11 x1 t05 02 gradient (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/18.jpg)
Two lines are parallel iff they have the same slope1 2i.e. m m
![Page 19: 11 x1 t05 02 gradient (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/19.jpg)
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
![Page 20: 11 x1 t05 02 gradient (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/20.jpg)
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
![Page 21: 11 x1 t05 02 gradient (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/21.jpg)
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)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/22.jpg)
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
![Page 23: 11 x1 t05 02 gradient (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/23.jpg)
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
![Page 24: 11 x1 t05 02 gradient (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/24.jpg)
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)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/25.jpg)
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)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/26.jpg)
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
![Page 27: 11 x1 t05 02 gradient (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/27.jpg)
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
![Page 28: 11 x1 t05 02 gradient (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/28.jpg)
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)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/29.jpg)
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)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/30.jpg)
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)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/31.jpg)
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
![Page 32: 11 x1 t05 02 gradient (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/32.jpg)
(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.
![Page 33: 11 x1 t05 02 gradient (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/33.jpg)
(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
![Page 34: 11 x1 t05 02 gradient (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/34.jpg)
(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
![Page 35: 11 x1 t05 02 gradient (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/35.jpg)
(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
![Page 36: 11 x1 t05 02 gradient (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/36.jpg)
(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
![Page 37: 11 x1 t05 02 gradient (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/37.jpg)
(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
![Page 38: 11 x1 t05 02 gradient (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/38.jpg)
To prove three points (A, B, C) are collinear;
![Page 39: 11 x1 t05 02 gradient (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/39.jpg)
To prove three points (A, B, C) are collinear;
( ) find ABi m
![Page 40: 11 x1 t05 02 gradient (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/40.jpg)
To prove three points (A, B, C) are collinear;
( ) find ABi m
( ) find ACii m
![Page 41: 11 x1 t05 02 gradient (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/41.jpg)
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
![Page 42: 11 x1 t05 02 gradient (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042817/55a5deff1a28ab88558b4739/html5/thumbnails/42.jpg)
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*