CHAPTER 7 COORDINATES

COORDINATES Mathematics
Form 2

Resources: CHAPTER 7: COORDINATES

Textbook (page 120 – 143)

Websites

etc

Learning standard:
7.1.1 Explain the meaning of distance between two points on the Cartesian plane.
7.1.2 Derive the formula of the distance between two points on the Cartesian plane.
7.1.3 Determine the distance between two points on the Cartesian plane.
7.1.4 Solve problems involving the distance between two points in the Cartesian coordinate system.

7.1 DISTANCE IN A CARTESIAN COORDINATE SYSTEM

y

6 x
5
4
3
2
1

O 1234 5

Figure 1 Figure 2

Figure 1 shows the plan of your classroom during the examination week. We can represent this
plan on a square of grid (Figure 2).

In Figure 2, mark the location of your seat with M and your best friend with F.
Points M and F shows the location of your seat and your best friend’s seat.

The coordinates of M are ( ___ , ___ ) and the coordinates of F are ( ___ , ___ )

Note:
Figure 2 is the first quadrant of Cartesian plane. The horizontal of number line is called the

____________ whereas the vertical number line is called the __________ . The intersection

point of the ___________ and ___________ is known as _____________, represent with

_____________

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COORDINATES Mathematics
Form 2

Activity 1:
1. Using the grid prepared, draw a Cartesian plane which is involved all the four quadrants.

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COORDINATES Mathematics
Form 2

2. Label x-axis, y-axis, the origin as O and the values on the coordinate axes is from
– 10 to 10. The scale of the both axes are 1 : 1.

3. Plot and label the following points.

Point Distance from the y-axis Distance from the x-axis
A(3, 0) 9 units 8 units

B(0, 4) 7 units

C(3, 4)

D(6, 8)

E(-5, 0)
F(–9, 7)
G(–7, 9)
H(–9, 10)
I(0, –9)
J(–3, –4)
K(–8, –7)
L(–5, –2)
M(–6, –8)
N(6, –7)
P(10, –1)
Q(8, –8)
R(3, –9)

4. Then, state their distances from the y-axis (column 2) and the x-axis (column 3) –
Refer to Example 2, on page 3
(Remark: The distances are always positive.)

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COORDINATES Mathematics
Form 2

Activity 1: Finding the distance between two points

A Both points have common y-coordinates

1. From the grid given, draw a Cartesian
plane which is involved all the four
quadrants.

2. Plot the points P(–4, –4) and Q(3, –4)
3. Both points have common

y-coordinates
4. Find the distance between P and Q

Answer:

B Both points have common x-coordinates

1. From the grid given, draw a Cartesian
plane which is involved all the four
quadrants.

2. Plot the points L(–3, –4) and M(–3, 3)
3. Both points have common

x-coordinates
4. Find the distance between L and M

Answer:

C Using Pythagoras’ Theorem

1. From the grid given, draw a Cartesian
plane which is involved all the four
quadrants.

2. Plot the points R(4, 4) and S(–2, –4)
3. Draw an appropriate right-angled

triangle using RS as hypotenuse
(page 13)
4. Find the distance between R and S

Answer:

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COORDINATES Mathematics
Form 2

Conclusion: y R(x2, y2)
y2
The distance between two points on a Cartesian plan
where P(x1, y1) and R(x2, y2) is given by

PR = (y2 - y1)

y1 P(x1, y1) (x2 - x1) Q

0 x1 x2 x

Problem Solving

Activity 3:

1. Given that A(–3, 1), B(–2, 3), C(3, 5) and D form a parallelogram, find the length of diagonal
BD

Answer:

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COORDINATES Mathematics
Form 2

2. Given that, the coordinates of J and K are (4, 5) and K(4, -3) respectively, find the distance
between J and K.

Answer:

3. Find the distance of the line which join the following pairs of points,
(a) P(-6, 8) , O(0, 0)
(b) R(2, 2), S(-1, -2)
(c) T(5, 0), U(0, -12)

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COORDINATES Mathematics
Form 2

Activity 4: Self Practice 7.1

Learning Standard
7.2.1 Explain the meaning of midpoint between two points on the Cartesian plane.
7.2.2 Derive the formula of the midpoint between two points on the Cartesian plane.
7.2.3 Determine the coordinates of midpoint between two points on the Cartesian plane.
7.2.4 Solve problems involving midpoint in the Cartesian

7.2 MIDPOINT IN THE CARTESIAN COORDINATE SYSTEM

Activity 1: Finding the midpoint between two points

A Both points have common y-coordinates

1. From the grid given, draw a Cartesian
plane which is involved all the four
quadrants.

2. Plot the points P(-4, -4) and Q(2, -4)
3. Both points have common

y-coordinates
4. Find the coordinates of the midpoint of

the line PQ

Answer:

The midpoint of PQ is ( ____, _____ )

B Both points have common x-coordinates

1. From the grid given, draw a Cartesian
plane which is involved all the four
quadrants.

2. Plot the points L(3, -4) and M(3, 2)
3. Both points have common

x-coordinates
4. Find the coordinates of the mid point of

the line LM

The midpoint of LM is ______

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COORDINATES Mathematics
Form 2

C Using the formula

1. From the grid given, draw a Cartesian
plane which is involved all the four
quadrants.

2. Plot the points R(1, 4) and S(-3, -4)
3. Draw an appropriate right-angled

triangle using RS as hypotenuse
(page 19)
4. Find the coordinates of the midpoint of
the line RS.

The midpoint of RS is ______

The formula of midpoint for (x1, y1) and (x2, y2) is  x1  x2 , y1  y2 
 2 2 

Activity 2:

Using the formula given, check the answer for activity 1.

A The coordinates of the midpoint of the line PQ:

  4  2 '  4   4 =
2 2

B The coordinates of the midpoint of the line LM:

C The coordinates of the midpoint of the line RS:
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COORDINATES Mathematics
Form 2

Enrichment 1 Activity 3: Self Practice 7.2
Activity 4: Self Practice 7.3 & Generating Excelence

1. From the grid given, draw a Cartesian plane which is involved all the four quadrants.

For questions, 2, 3 & 4, please refer to the Cartesian plane above:

2. Plot the following points:

P(2, 5) Q(-1, 1), R(-7, -3) S(3, -3) T(2, -6) L(8, -6)

3. Find the distance of each of the following:

(i) RS = _______ units (iii) PQ = _______ units
(ii) PT = _______ units (iv) L from the origin = _______ units

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COORDINATES Mathematics
Form 2

4. Find the midpoint between the two points given:

(i) R and S = ________ (ii) Q and L = ________

5. Refer to the figure as shown and find D(-8, 5) y
(a) the coordinates of the midpoint of the line DB B(0,3)
(b) AB

0 x

A(-6, -5)

y

6. In the figure, N is the midpoint of straight line LM. Find M(5, 48)
(a) x-coordinates of L. N(15, 24)
(b) MN x
L
0 [5 marks]

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COORDINATES Mathematics
Form 2

ENRICHMENT 2

TICKET 7(a) TICKET7(b)
Based on the diagram, state the
coordinates of the points given. Plot each of the following points in the

y Cartesian plane given:

8 (a) P(2, 0)

N 6 (b) Q(5,2) y
(c) R(4, 7)
4 Q
R (d) S(0, 3) 8
(e) T(4, –2) 6
4
(f) M(–6, – 1)
(g) N(–2, – 8)
(h) V(4, –5)

M 2 2

P

–6 –4 –2 0 2 S x –6 –4 –2 0 2 4

4

6 – 2 T 6 –2

Y

–4 –4
–6
U –8
X – 6

W
– 8 V

TICKET 7(c) TICKET 7(d)
1. State five points on the same straight
1. Find the distance between each of the
line which parallel to y-axis.
following pairs of points:
(a) P(3, 4) and T(–3, 4) 2. Find the distance between the origin and
(b) Q(–3, –5) and U(3, –5) the following points:
(c) R(–2, 6)) and V(1, 6) (a) P(–3, 4)
(d) S( 3, –4) and W(–5, –4) (b) Q(5, –12)

2. Find the distance of the following points 3. Find the distance between each of the
following pairs of points.
from the origin. (a) P(– 3, 2) and Q(–6, 6)
(b) R(3 ,– 6) and S( 15, –3)
(a) P(5, 12)
(b) Q(–7, 24) 4. The distance between points A(k, 5) and
B(2, 5) is 4 units. Find the possible
(c) R(8, 15)) values of k.
(d) S( 3, –4)
5. Given that the distance between points
3. The distance between points A(2, k) and P(–3, –4) and Q(–3, n) is 6 units, find
the possible values of n.
B(2, 3) is 2 units. Find the possible

values of k.

4. Given that the distance between points
P(–3, –4) and Q(n, –4) is 6 units, find 11

thepossible values of n.

COORDINATES Mathematics
Form 2

TICKET 7(e) TICKET 7(f)
1. State five points on the same straight 1. Find the coordinates of the midpoint of

line which is parallel to y-axis. a straight line joining each of the
following pairs of points.
2. Find the distance between the origin and (a) P( 3, –2) and Q( –6, –2)
the following points: (b) R(–6, –1) and S(10, –1)
(a) R(8, 15) (c) T(–7, 5) and U(–7, –8)
(b) S(7, 24)
2. Find the coordinates of the midpoint of
3. Find the distance between each of the a straight line joining each of the
following pairs of points. following pairs of points.
(a) T(8, –5) and U(–1, 7) (a) P( 3, –2) and Q( –6, 8)
(b) V(–8, 10) and W(–2, 2) (b) R(–6, –1) and S(10, 5)
(c) X( – 12, –7) and Y(– 3, 5) (c) T(–7, 5) and U(–6, –8)

4. State five points on the same straight 3. M(–10, 12) is the midpoint of a straight
line which parallel to x-axis. line joining points P(–5, h) and Q(k, 2).
Find the value of h and k.
5. Find the distance between each of the
following pairs of points. 4. The distance between points P(– 6 , 2)
and Q(–6, y) is 5 units. Find the possible
(a) P(7, – 4) and Q(2, 8) values of y.
(b) R(1, 6) and S( 4, 2) .

TICKET7(g) TICKET 7(h)
1. Find the coordinates of the midpoint of 1. The distance between point A(2, k) and

a straight line joining each of the B(2, 3) is 2 units. Find the possible
following pairs of points:
(a) (4, 2) and (4, – 8) values of k.
(b) (–3, –6) and (9, –6) 2. Point K(– 3, 1), L(2, 3) and N(2, y) form

2. Find the coordinates of the midpoint of an isosceles triangle. LMN is a straight
a straight line joining points: line.
(a) A(2, –3) and B(4, 5)
(b) (–2, 1) and ( –6, 9) L

3. M is the midpoint of the straight line KM
KL. The coordinates of points K and M
are (4, 5) and (3, – 6) respectively. Find 3. Find the value of yb.etweNen J(7, – 2) and
the coordinates of point L. Find the distance

4. Given that M(– 6, 2) is the midpoint of K(– 1, – 17)
the straight line joining points A(a, 5)
and B(– 4, b). Find the values of a and 12 4. Given that the distance between points
b. P(–3, –4) and Q(–3, n) is 6 units, find
the possible values of n.
5. M(5, – 2) is the midpoint of the straight
line joining point A(x, – 10) and B(8, y). 5. The distance between points P(– 6 , 2)
and Q(x, 2) is 10 units. Find the possible