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Question (1):
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]
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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). |
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Question (2):
Show that the line segment joining the points (-5,8) and (10,-4) is trisected by co-
ordinate axes. [3 Mark]
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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. |
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Question (3):
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]
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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).
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Question (4):
Prove that for the vertices A (x1,y1), B (x2,y2) and C (x3,y3) of a triangle ABC, its
centroid is  [3 Mark]
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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
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Question (5):
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]
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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
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Question (6):
Two vertices of a triangle are (3,-5) and (-7,4). Find the third vertex, given that the
centroid is (2,-1). [3 Mark]
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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). |
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Question (7):
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]
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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.
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Question (8):
In what ratio does the point  divide the line segment joining the points (3,-5)
and (-7,9)? [2 Mark]
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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
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Question (9):
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.
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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
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Question (10):
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.
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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
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Question (11):
Show by section formula, that the points (3,-2), (5,2) and (8,8) are collinear.
[2 Mark]
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Answer:
Let A, B, C be the points whose co-ordinates are (3,-2), (5,2) and (8,8) respectively.
Any point on AC, dividing the segment in the ratio  :1 is
If this point is B, then
8 + 3 = 5 + 5 and 8 - 2 = 2 + 2
Solving both the equations, we get  that is B divides AC in the ratio 2:3.
 B lies on the line AC.
Hence, the three given points A, B and C are collinear.
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Question (12):
If the points (-2,-1), (1, 0), (x, 3), (1, y) form a parallelogram, the find the value of x,
y. [2 Mark]
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Answer:
The diagonals of a parallelogram bisect each other. That is,
The midpoint of AC = The midpoint of BD
That is 
 (x, y) = (4, 2) |
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Question (13):
Find the co-ordinates of the point dividing the line segment joining the points:
[Each 2 Mark]
(i) A(5, 7), B(2, 4) in the ratio 2:1.
(ii) A(-6, 6), B(-1, 6) in the ratio 2:3
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Answer:
(i) Let P(x, y) be the point which divides AB in the ratio 2:1.
Then co-ordinates of P are given by
m = 2 and n = 1,
 Coordinates of P(x, y) = P(3, 5).
(ii) A(-6, 6), B(-1, 6) in the ratio 2:3
Let point P(x, y) divide the line joining points A(x1, y1), B(x2, y2) in the ratio 2:3
x1 = -1y2 = 6
m = 2, n = 3
Then P(x, y) is given by
x = -4 y = 6
 Coordinates of P(-4, 6).
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Question (14):
Find the coordinates of the mid points of the line segment joining the points:
[Each 2 Mark]
(i) (8, -3), (-4, -7)
(ii) (4b, 2b), (b, 6b)
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Answer:
If A(x1, y1), B(x2, y2) is the end points of line segment, then mid point P(x, y) is given
by the formula,
P(x, y) = (2, -5)
(ii) (4b, 2b), (b, 6b)
Let x1 = 4b y1 = 2b
x2 = b y2 = 6b
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Question (15):
In what ratio does the point P(8, 4) divide the join of A(5, -2) and B(9, 6)?
[2 Mark]
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Answer:
Let the point P(8, 4) divide the line joining Point A(5, -2) and B(9, 6) in the ratio m:n.
3n = m 6n = 2m
m:n = 3:1
P divides line AB in the ratio 3:1.
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Question (16):
In what ratio does the point C(-3, 0) divide the join of A(-7, 2) and
B(-1, -1)? [2 Mark]
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Answer:
Let the point C(-3, 0) divide the line joining A(-7, 2) and (-1, -1) in the ratio m:n.
Then according to section formula,
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Question (17):
In what ratio is the line segment joining the points (5, -4) and (-3, 2) divided by X-
axis? [2 Mark]
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Answer:
Let the point C on X-axis divide the line joining AB in the ratio m:n
Hence its coordinates will be (x, 0), coordinates of C are given by,
0 = 2m - 4n
m:n = 2:1 |
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Question (18):
In what ratio is the line segment joining the points (-2, -3) and (5, 4) divided by y -
axis? Also write the coordinates of the points of intersection on y - axis. [2 Mark]
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Answer:
Let the Point on y - axis C(0, y) divide the line joining A(-2, -3) and B(5, 4) in ratio m:n.
By section formula,
 Hence y - axis divides the line segment in the ratio 2:5.
(as multiplication is commutative)
 (0, 1) is the point on y - axis. |
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Question (19):
In what ratio is the line segment joining the points (9, 2) and (-3, -2) divided by y -
axis? Also write the coordinates of the point of intersection. [3 Mark]
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Answer:
Let A(9, 2), B(-3, -2)
given the line joining the points AB is divided by y - axis, Let it be in the ratio m:n.
Let that point Xn be P(x, y) then
P(x, y) lies on y - axis, hence x = 0, P(0, y)
 
0 = -3m + 9n
3m = 9n
m = 3n
m:n = 3:1
 add 1 to both the sides,
 Point of intersection = (0, -1)
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Question (20):
Show that the co-ordinates of the point dividing the join of the points (3, -5) and (-4,
6) in the ratio of 2:3 and the join of points (-4, 6) and (3, -5) in the ratio of 3:2 are the same.
[3 Mark]
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Answer:
Let P(x, y) be the point dividing the join points A(3, -5) and (-4, 6) in the ratio 2:3.
According to section formula,
Let Q (a, b) be the point which divides the line joining the points in the ratio 3:2.
Let m = 3, n= 2, points are (-4, 6), (3, -5).
By section formula,
Hence the coordinates of point P(x, y) = Point Q
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Question (21):
P(5, 3) is the mid point of the line joining the points A(12, 9) and B(x, y). Find the
value of x and y. [2 Mark]
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Answer:
A(12, 9), B(x , y), given P(5, 3) is the mid point of AB
Mid point is given by
Hence Coordinates of B are (-2, -3). |
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Question (22):
Find the coordinates of the mid points of the sides of a triangle whose vertices are
A(3, -4), B(-7, 6), C(-5, -2). [3 Mark]
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Answer:
Let P(x1, y1), Q(x2, y2), R(x3, y3) be the coordinates of AB, BC and CA.
By mid point formula,
= (-2, 1)
 Mid point of AB, P (x1, y1) = (-2, 1)
= (-6, 2)
 Mid point of BC, Q (x2, y2) = (-6, 2)
= (-1, -3)
Hence the mid points are P(-2, 1), Q(-6, 2) and R(-1, -3). |
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Question (23):
The points A(9, 6), B(-11, 8) and C(-3, 4) are the vertices of a triangle ABC. AD is the
median. Find the Co-ordinates of D. Hence find the Coordinates of the centroid. [3 Mark]
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Answer:
AD is the median  The line joining the vertice A to the mid point of the of the opposite side.
Hence D is the mid point of BC.
= (-7, 6)
Centroid G is a point which is formed by the intersection of three medians. It divides the median in the ratio 2:1(from the vertex).
Hence Coordinates of G(a, b) is given by the section formula,
Let m = 2, n = 1
x1 = 9, y1 = 6
x2 = -7, y2 = 6
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Question (24):
The points (1, 1), (9, 3), (11, 8) and (3, 6) are the vertices of a Quadrilateral. Show
that the diagonals bisect each other. Hence classify the Quadrilateral. [3 Mark]
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Answer:
Let A(1, 1), B(9, 3), C(11, 8), D(3, 6) are the vertices of the Quadrilateral.
AC and BD are the diagonals. If they bisect each other, then O(x, y) is the mid point of AC and
BD.  DO = OB, AO = OC.
Coordinates of O(x, y) if it is the mid point of AC
Coordinates of O(x, y), if it is the mid point of
As the values for both the mid points are same they coincide.  O is the point of bisectors of
diagonals. Hence diagonals bisect each other. Therefore it is a parallelogram. |
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Question (25):
In the circle with centre O, CD is a diameter. If the Coordinates of C and D are (10, 0)
and (-2, -3). Find the Coordinates of the centre O. [2 Mark]
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Answer:
CD is the diameter.
O is the centre of circle. Diameter = 2 x radius
Hence O is the mid point of the diameter.
According to Co-ordinates of mid point formula,
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