form 4 geometry coordinate.docx

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FORM 4 (Chapter 6) : Coordinate Geometry

Evaluate Yourself

1. Calculate the distance between the origin O( 0,0 ) and point A( 6,8 ).

2.A(1,7)

C

B

D(9,1)

In the diagram, C is the midpoint of AD and B is the midpoint of CD.

Find the coordinates of point B.

3. State the modulus value of each of the following.

(a) |-6|

(b) |- |

(c) |-2.8|

4. Describe the locus of the moving point

(a) P which is always 5 cm away from the fixed point O,

(b) T which is always at the same distance from points P

and Q.

6.1 Distance Between Two Points

The distance between points ( ) and is

√



Example 1

Find the distance between the points A( 2, -2 ) and B( -4, -5 ).

Example 2

Find the possible values of p if the distance between two points A(1, 2 ) and B( p, 14 ) is 13.

6.2 Division of a Line Segment

6.2a Midpoints

The midpoint of the points (

) and

is

Example 3

If M ( -3,

is the midpoint of the points P( -4, -7 ) and Q, find

the coordinates of the point Q.

SPM (1) 2000

Given that the points E( 5, 6 ), F( 8, 7 ), g( 7, 3 ) and H are the four

vertices of a parallelogram, find the coordinate of point H.

6.2b Point that Internally divides a Line Segment in the Ratio m : n

,

Example 4



The point T internally divides the line segment joining the points

P(3, -3 ) and Q( -2, 4 ) in the ratio 2 : 5. Find the coordinates of point

T.

Example 5

N is a point on the line segment joining Q( -1, -3 ) and R( 4, 1 ) suchthat QN = 4NR. Find the coordinates of point N.

Example 6

The point H( 1, -1 ) internally divides the line segment joining

points A( -2, 2 ) and B in the ratio 3 : 2. Find the coordinates of point

B.

SPM (2) 2003

The coordinates of the points K and M are ( 5, 4 ) and ( 1, -2 )

respectively. If the point L( , k ) lies on the straight line KM, find

(a) The ratio KL : LM, (b) the value of k.

SPM (3) 2003

The points A(3m, 2n ), B( m, n ) and C( p, 2p ) lie on a straight line.

The point B internally divides the line segment AC in the ratio 5 : 2.

Express m in terms of n.

6.3 Areas of Polygons

6.3a To Find the Area of a Triangle based on the Areas of Specific Geometrical Shapes

Example 7

The points A( 4, -2 ), B( 1, -1 ) and C( -1, 3 ) from a triangle. Find the

area of ABC based on the area of specific geometrical shapes.6.3b To Find Areas of Polygons Using Formulae



Example 8

Calculate the area of ABC with the vertices A( 1, 3 ), B( 5, 1 ) and

C( 6, 7 ).

Example 9

Calculate the area of quadrilateral TUVW with the vertices T(-2, 13),

U( 10. 12 ), V( 2, 3 ) and W( -10, 4 ).

Example 10

Find the values of p if the points A( 2, 1 ), B( 6, p ) and C( 3 p, ) are

collinear.

Example 11

PQR has the vertices P( 3, 4 ), Q( 5, -3) and R( 7, 3 ).

(a) Calculate

( i ) the area of PQR,

( ii ) the distance of PQ.

(b) Hence, find the perpendicular distance from point R to PQ.

6.4 Equations of Straight Lines

6.4a Axes Intercepts and Gradients

( )

Example 12

Given the points P( 3, 4 ) and Q( -4, 2 ), find the gradient of the

straight line PQ.



Example 13

The -intercept and -intercept of the straight line PQ are 3 and -9

respectively. Find the gradient of PQ.

Example 14

Find the values of k if the points R( -3, -2 ), S( 1, k +1 ) and T(3, 2k +1)

are collinear.

6.4b Equations of Straight Lines

1. Case 1 The gradient and the coordinates of a point are given

The equation of the straight line is

Case 2 The coordinates of two points are given


Case 3 The -intercept and -intercept are given




2. The equation of a straight line can be expressed in three forms :

Equation in the gradient form

Equation in the general form

Equation in the intercept form

Example 15

Find the equation of a straight line that passes through the point (3, -2) and

has a gradient of .

Example 16

Find the equation of a straight line that passes through the points

(-2, -3) and ( -4, 7 ).

Example 17

The -intercept and the -intercept of the straight line MN are 4 and

3 respectively. State the equation of the straight line MN in theintercept form.



Example 18

Find the equation of the straight line that passes through the points

R( 3, 5 ) and S( 6, 4 ), expressing your answer in each of the

following forms.

(a) Gradient form (c) intercept form(b) General form

Example 19

Find the gradient and the -intercept of the straight line that has the

equation 6.4c Intersection Points

Example 20

Point T is the intersection point of the straight lines

and The coordinates of point U are ( 3, 2 ). Find the

equation of the straight line TU.

6.4d Further Examples on Equations of Straight Lines

SPM (4) 2002

y

A(-2,2)

C

0 x

B(-3,-5)

The diagram shows ABC with an area of 18 square units. The

equation of the straight line BC is Find the

coordinates of point C.



6.5 Parallel Lines and Perpendicular Lines

6.5a Parallel Lines

Example 21

Given the points G( -3, -1 ), H( 0, 8 ), T( 1, 4 ) and U( 5, 16 ), show

that GH is parallel to TU.

Example 22

Find the equation of the straight line that passes through the point

( -1, 3 ) and is parallel to the straight line

6.5b Perpendicular Lines

Example 23

Given the points A( -2, 4 ), B( 4, 2 ), P( -1, -3 ) and Q( 2, 6 ), show

that AB is perpendicular to PQ.

SPM (5) 2003

The straight lines PQ and RS are perpendicular to each other. If the

equation of PQ and RS are and

respectively, find the value of

Example 24

The coordinates of points A, B, P and Q are ( 2, 1 ), ( 4, 9 ), ( 1, 6 ),

and ( 6, -1 ) respectively. Find the equation of the straight line that

passes through the midpoint of AB and is perpendicular to the

straight line PQ.



SPM (6) 2004

y

L(-4,5)

M(0,1)0 x

N

In the above diagram, point M( 0, 1 ) internally divides the line

segment joining the points L( -4, 5 ) and N in the ratio 2 : 3. Find

(a) The coordinates of point N,

(b) The equation of the straight line that is perpendicular to the

straight line LN and passes through the point N.

Example 25

The coordinates of the points P and Q are ( -3, 1 ) and ( 5, 11 )

respectively. Find the equation of the perpendicular bisector of PQ.

SPM (7) 2001

In the above diagram, the points P( 2, 7 ), Q( 9, 5 ) and R ( 5, 3 ) are

the midpoints of the straight lines JK, KL and LJ respectively. Find

(a) The equation of the straight line KL,

(b)

The equation of the perpendicular bisector of JL.

J

K

L

P(2,7)Q(9,5)

R(5,3)



6.5c Further Examples on Parallel Lines and Perpendicular

Lines

Example 26

In the above diagram, ABCD is a trapezium. The point D lies on the

-

), where is a constant, find

(a) The equation of the straight line CD,

(b) The value of

B(0,6)

SPM (8) 2004

In the above diagram, the straight lines AB and CD are

perpendicular to the straight line BC. Find the equation of the

straight line CD in the intercept form.

C(2,5)

B(5,3)

A(4,k)

D

A(-3,0) O

C

D



Example 27

In the above diagram, the points P( 6, 7 ) and R( 10, -1 ) are the

opposite vertices of a rhombus PQRS. The point Q lies on the

-axis

and T is the midpoint of PR. Find

(a) The equation of the straight line QTS,

(b) The coordinates of point Q,

(c) The area of rhombus PQRS.

SPM (9) 2001

The straight line cuts the -axis and the -axis at points

A and B respectively. A point C is such that the gradient of the

straight line CB is 2 and the straight line AC is perpendicular to the

straight line CB. Find

(a) The equation of the straight line CB,

(b) The equation of the straight line AC,(c) The coordinates of point C.

P(6,7)

R (10,-1)

T

S

Q



6.6 Equation of a Locus that Involves Distance Between Two

Points

6.6a Equations of Loci that Involve Distance Between Two

Points

Example 28

Find the equation of the locus of a moving point P such that its

distance from the point A( 2, 4 ) is 2 units.

Example 29

Find the equation of the locus moving point Q such that its distancefrom the points B( 3, -7 ) and C( -5, 1 ) are equal.

Example 30

P( 2, 0 ) and Q( 0, -2 ) are two fixed points. The point T moves in

such a way that PT : TQ = 1 : 2. Find the equation of the locus of

point T.

6.6b Further Examples on the Equation of Loci

SPM (10) 2004

If the points P( 0, 8 ), Q( 8, 0 ) and R( ) lie on the circumference

of a circle with a diameter PQ, find the equation of the locus of the

moving point R.

SPM (11) 2001

Given the points A( 0, 1 ) and B( 6, 4 ), find the equation of the locus

of a moving point P such that the triangle APB always has a right

angle at P.



SPM (12) 2002

A point Q moves in such a way that its distance from the point T( 2, -

3 ) is twice its distance from the point S( 0, 3 ).

(a) Find the equation of the locus of point Q.

(b) Hence, determine whether the locus of point Q intersects the-axis or not.

SPM (13) 2001

Point P( ) moves in such a way that its distance from point Q( 3,

1 ) is always two times its perpendicular distance from the straight

line Find the equation of the locus of the moving point P.

SPM (14) 2003

Point R moves along the circumference of a circle, with centre M( 1,

3 ). The circumference of the circle passes through points A( -3, 0 )

and B( ).

(a) Find

( i ) the equation of the locus of the point R,

( ii ) the values of (b) The tangent to the circle at the point A cuts the -axis at

point C. Find the area of the triangle OAC.

form 4 geometry coordinate.docx

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