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horizontal runm

horizontal runm

y

x

horizontal runm

y

x

l 5,5 2,1

horizontal runm

y

x

l 5,5 2,1

2 1

2 1

(2) y ymx x

horizontal runm

y

x

l 5,5 2,1

2 1

2 1

(2) y ymx x

5 15 2l

m

horizontal runm

y

x

l 5,5 2,1

2 1

2 1

(2) y ymx x

5 15 2l

m 43

horizontal runm

y

x

l 5,5 2,1

2 1

2 1

(2) y ymx x

5 15 2l

m 43

L

horizontal runm

y

x

l 5,5 2,1

2 1

2 1

(2) y ymx x

5 15 2l

m 43

L

120

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

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

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

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

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

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

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

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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