Question 1
Question: Points A, B, C and D divides the line segment joining the points (5,-10) and the origin in five equal parts. Find the co-ordinates of A, B, C and D. [5 Mark]
Answer: 
In the figure, D divides PQ in the ratio 1:4.
D is (1, -2).
C divides PQ in the ratio 2:3.
C is (2,-4).
B divides PQ in the ratio 3:2.
B is (3,-6).
A divides PQ in the ratio 4:1.
A is (4,-8).
Question 2
Question: Show that the line segment joining the points (-5,8) and (10,-4) is trisected by co- ordinate axes. [3 Mark]
Answer: 
Let P and Q be the two points which divide AB in the ratio 1:2 and
2:1 respectively.
The co-ordinates of P are given by
P (0,4) is a point on the y-axis since x-co-ordinate is zero.
The co-ordinates of Q are
= (5,0)
Q (5,0) is a point on the x-axis since y-co-ordinate is 0.
The points of trisection of AB lie on the co-ordinate axes.
Question 3
Question: In the given figure, P(3,1) is a point on the line segment AB such that AP:PB = 2:3. Find the co-ordinates of A and B. [3 Mark]
Answer: 
Since A is a point on x-axis, let the co-ordinates of A be (x,0).
Since B is a point on y-axis, let the co-ordinates of B be (0,y).
The co-ordinates of A are (5,0).
Question 4
Question: Prove that for the vertices A (x1,y1), B (x2,y2) and C (x3,y3) of a triangle ABC, its centroid is 
[3 Mark]
Answer: 
ABC is the given triangle in which AD is the median of BC.
D divides BC in the ratio 1:1
(D is the midpoint of BC)
The co-ordinates of D are given by
That is 
The centroid G divides AD in the ratio 2:1, therefore the co-ordinates of G are given by
Question 5
Question: Find the co-ordinates of the point of intersection of the medians of triangle ABC, given A = (-2,3), B = (6,7), C = (4,1). [3 Mark]
Answer: 
In 
ABC, AD is the median.
D is the midpoint of BC
Let G (x,y) be the point of intersection of the medians (centroid), then G divides AD in the ratio 2:1.
The co-ordinates of G are
Question 6
Question: Two vertices of a triangle are (3,-5) and (-7,4). Find the third vertex, given that the centroid is (2,-1). [3 Mark]
Answer: 
Since D is the midpoint of BC, the co-ordinates of D are given by 
Since the centroid G, divides the median in the ratio 2:1,
Using section formula, we have
The co-ordinates of the vertex A are (10,-2).
Question 7
Question: The line segment joining A(2,3) and B(6,-5) is intersected by the x-axis at point K. Write down the ordinate of K. Hence find the ratio in which K divides AB. [3 Mark]
Answer: 
The point K is on the x-axis.
Therefore, the ordinate of K is 0.
Let the co-ordinates of K be (x,0) and K divides AB in the ratios 
:1.
Using section formula, we have
K divides AB in the ratio 3:5.
Question 8
Question: In what ratio does the point 
divide the line segment joining the points (3,-5) and (-7,9)? [2 Mark]
Answer: 
Let the point 
divide the line segment joining A(3,-5) and
B (-7,9) in the ratio k:1.
From (1), we have
k+1 = 6-14k
15 k = 5
Question 9
Question: In the given figure, the points P and Q have co-ordinates (3,4) and (0,2) respectively. [3 Mark]
Find
(i) Co-ordinates of R and
(ii) The area of quadrilateral OMPQ.
Answer: Let Q divide RP in the ratio 
:1 and the co-ordinates of R are (x,0). Then, using section formula for y-co-ordinate of Q, we have
= 1
Q divides RP in the ratio 1:1.
(That is Q is the midpoint of RP)
The co-ordinates of R are (-3,0).
The area of the trapezium OMPQ
Question 10
Question: Given O(0,0), P(1,2), S(-3,0). P divides OQ in the ratio 2:3 and OPRS is a parallelogram. Find [3 Mark]
(i) the co-ordinates of Q.
(ii) the co-ordinates of R and
(iii) the ratio in which RQ is divided by the y-axis.
Answer: (i) 
Let the co-ordinates of Q be (x,y), then
(ii) Since OPRS is a parallelogram, the diagonals bisect each other.
That is the midpoint of RO and mid point of PS coincide.
The midpoint of RO = The midpoint of PS
Let the co-ordinates of R be (x,y), then
x = -2, y = 2
The co-ordinates of R = (-2,2).
(iii) Let L(0,y) be a point on the y-axis which dives RQ in the ratio 
:1, then
