1

On which axis do the following points lie?

(i) P (5, 0) (ii) Q (0 -2)

(iii) R (- 4, 0) (iv) S (0, 5)

view answer >
2

Let ABCD be a square of side 2a. Find the coordinates of the vertices of this square when

(i) A coincides with the origin and AB and AD are along OX and OY respectively.

(ii) The centre of the square is at the origin and coordinate axes are parallel to the sides AB and AD respectively.

view answer >
3

The base PQ of two equilateral triangles PQR and PQR' with side 2a lies along y-axis such that the mid-point of PQ is at the origin. Find the coordinates of the vertices R and R' of the triangles.

view answer >
1

Find the distance between the following pair of points:

(i) (-6,7) and (-1, -5)

(ii) (a + b, b + c) and (a - b, c - b)

(iii) (a sin a, - b cos a) and (-a cos a, b sin a)

(iv) (a, 0) and (0, b)

view answer >
2

Find the value of a when the distance between the points (3, a) and (4, 1) is .

view answer >
3

If the points (2, 1) and (1,-2) are equidistant from the point (x, y), show that x + 3y = 0.

view answer >
4

Find the values of x, y if the distances of the point (x, y) from (-3, 0) as well as from (3, 0) are 4.

view answer >
5

The length of a line segment is of 10 units and the coordinates of one end-point are (2,-3). If the abscissa of the other end is 10, find the ordinate of the other end.

view answer >
6

Show that the points A(- 4, -1), B(-2, - 4), C(4, 0) and D(2, 3) are the vertices points of a rectangle.

view answer >
7

Show that the points A (1,- 2), B (3, 6), C (5, 10) and D (3, 2) are the vertices of a parallelogram.

view answer >
8

Prove that the points A (1, 7), B (4, 2), C (-1, -1) and D (-4, 4) are the vertices of a square.

view answer >
9

Prove that the points (3, 0), (6, 4) and (- 1, 3) are vertices of a right-angled isosceles triangle.

view answer >
10

Prove that (2, -2), (-2, 1) and (5, 2) are the vertices of a right angled triangle. Find the area of the triangle and the length of the hypotenuse.

view answer >
11

Prove that the points (2 a, 4 a), (2 a, 6 a) and are the vertices of an equilateral triangle.

view answer >
12

Prove that the points (2, 3), (-4, -6) and (1, 3/2) do not form a triangle.

view answer >
13

An equilateral triangle has two vertices at the points (3, 4) and (-2, 3), find the coordinates of the third vertex.

view answer >
14

Show that the quadrilateral whose vertices are (2, -1), (3, 4), (-2, 3) and (-3, -2) is a rhombus.

view answer >
15

Two vertices of an isosceles triangle are (2, 0) and (2, 5). Find the third vertex if the length of the equal sides is 3.

view answer >
16

Which point on x-axis is equidistant from (5, 9) and (- 4, 6)?

view answer >
17

Prove that the points (- 2, 5), (0, 1) and (2, - 3) are collinear.

view answer >
18

The coordinates of the point P are (-3, 2). Find the coordinates of the point Q which lies on the line joining P and origin such that OP = OQ.

view answer >
19

Which point on y-axis is equidistant from (2, 3) and (-4, 1)?

view answer >
20

The three vertices of a parallelogram are (3, 4), (3, 8) and (9, 8). Find the fourth vertex.

view answer >
21

Find the circumcentre of the triangle whose vertices are (-2, -3), (- 1, 0), (7, - 6).

view answer >
22

Find the angle subtended at the origin by the line segment whose end points are (0,100) and (10, 0).

view answer >
23

Find the centre of the circle passing through (2, 1), (5, - 8) and (2, - 9).

view answer >
24

Find the value of k, if the point P (0, 2) is equidistant from (3, k) and (k, 5).

view answer >
25

If two opposite vertices of a square are (5, 4) and (1, -6), find the coordinates of its remaining two vertices.

view answer >
26

Show that the points (-3, 2), (-5, -5), (2, -3) and (4, 4) are the vertices of a rhombus. Find the area of this rhombus.

view answer >
27

Find the coordinates of the circumcentre of the triangle whose vertices are (3, 0), (-1, -6) and (4,-1). Also, find its circumradius.

view answer >
28

Find a point on the x-axis which is equidistant from the points (7, 6) and (-3, 4).

view answer >
29

Show that the points A(5, 6), B (1, 5), C(2, 1) and D(6, 2) are the vertices of a square.

view answer >
29

Prove that the points A (2, 3), B (-2, 2), C (-1, -2), and D (3, -1) are the vertices of a square ABCD.

view answer >
30

Find the point on x-axis which is equidistant from the points (-2, 5) and (2,-3).

view answer >
31

Find the value of x such that where the coordinates of P, Q and R are (6,-1), (1, 3) and (x, 8) respectively.

view answer >
32

Prove that the points (0, 0), (5, 5) and (-5, 5) are the vertices of a right isosceles triangle.

view answer >
33

If the point P(x, y) is equidistant from the points A(5, 1) and B (1, 5), prove that x = y.

view answer >
34

Q (0, 1) is equidistant from P (5, -3) and R (x, 6), find the values of x. Also, find the distances QR and PR.

view answer >
35

Find the values of y for which the distance between the points P (2, -3) and Q (10, y) is 10 units.

view answer >
36

Find the centre of the circle passing through (6, -6), (3, -7) and (3, 3).

view answer >
37

Two opposite vertices of a square are (-1, 2) and (3, 2). Find the coordinates of other two vertices.

view answer >
38

Name the quadrilateral formed, if any, by the following points, and give reasons for your answers:

(i) A (-1, - 2), B (1, 0), C (-1, 2), D (-3, 0)

(ii) A (-3, 5), B (3, 1), C (0, 3), D (-1, - 4)

(iii) A (4, 5), B (7, 6), C (4, 3), D (1, 2)

view answer >
39

Find the equation of the perpendicular bisector of the line segment joining points (7, 1) and (3, 5).

view answer >
40

Prove that the points (3, 0), (4, 5), (-1, 4) and (-2, -1), taken in order, form a rhombus. Also, find its area.

view answer >
41

In the seating arrangement of desks in a classroom three students Rohini, Sandhya and Bina are seated at A (3, 1), B (6, 4) and C (8, 6). Do you think they are seated in a line?

view answer >
42

Find a point on y-axis which is equidistant from the points (5, - 2) and (- 3, 2).

view answer >
43

Find a relation between x and y such that the point (x, y) is equidistant from the points (3, 6) and (-3, 4).

view answer >
44

If a point A (0, 2) is equidistant from the points B (3, p) and C (p, 5), then find the value of p.

view answer >
45

Prove that the points (7, 10), (-2, 5) and (3, -4) are the vertices of an isosceles right triangle.

view answer >
46

If the point P (x, 3) is equidistant from the points A (7,-1) and B (6, 8), find the value of x and find the distance AP.

view answer >
47

If A (3, y) is equidistant from points P (8, -3) and Q (7,6) , find the value of y and find the distance AQ.

view answer >
48

If (0, - 3) and (0, 3) are the two vertices of an equilateral triangle, find the coordinates of its third vertex.

view answer >
49

If the point P (2, 2) is equidistant from the points A (-2, k) and B (-2k, -3), find k. Also, find the length of AP.

view answer >
50

If the point A (0, 2) is equidistant from the points B (3, p) and C (p, 5) the length of AB.

view answer >
51

If the point P (k -1, 2) is equidistant from the points A (3, k) and B (k,5), find the value of k.

view answer >
1

Find the coordinates of the point which divides the line segment joining (- 1, 3) and (4, -7) internally in the ratio 3 : 4.

view answer >
2

Find the points of trisection of the line segment joining the points:

(i) (5, -6) and (- 7, 5), (ii) (3, -2) and (-3, -4), (iii) (2, -2) and (-7, 4)

view answer >
3

Find the coordinates of the point where the diagonals of the parallelogram formed by joining the points (-2, -1), (1, 0), (4, 3) and (1, 2) meet.

view answer >
4

Prove that the points (3, -2), (4, 0), (6, -3) and (5, -5) are the vertices of a parallelogram.

view answer >
5

Three consecutive vertices of a parallelogram are (-2, -1), (1, 0) and (4, 3). Find the fourth vertex

view answer >
6

The points (3, -4) and (-6, 2) are the extremities of a diagonal of a parallelogram. If the third vertex is (-1, -3). Find the coordinates of the fourth vertex.

view answer >
7

Find the ratio in which the point (2,y) divides the line segment joining the points A (-2, 2) and B ( 3, 7). Also, find the value of y.

view answer >
8

If A (-1, 3), B (1, -1) and C (5, 1) are the vertices of a triangle ABC, find the length of the median through A.

view answer >
9

If the coordinates of the mid-points of the sides of a triangle are (1, 1), (2, -3) and (3, 4), find the vertices of the triangle.

view answer >
10

If a vertex of a triangle be (1, 1) and the middle points of the sides through it be (-2, 3) and (5, 2), find the other vertices.

view answer >
11

In what ratio is the line segment joining the points (-2, -3) and (3, 7) divided by the y-axis? Also, find the coordinates of the point of division.

view answer >
11

In what ratio is the line segment joining (-3, -1) and (-8, -9) divided at the point (-5, -21 /5)?

view answer >
12

If the mid-point of the line joining (3, 4) and (k, 7) is (x, y) and 2x + 2y + 1 = 0, find the value of k.

view answer >
13

Determine the ratio in which the straight line x - y - 2 = 0 divides the line segment joining (3, -1) and (8, 9).

view answer >
14

Find the ratio in which the line segment joining (-2, -3) and (5, 6) is divided by (i) x-axis (ii) y-axis. Also, find the coordinates of the point of division in each case.

(i) x-axis

view answer >
15

Prove that the points (4, 5), (7, 6), (6, 3), (3, 2) are the vertices of a parallelogram. Is it a rectangle.

view answer >
16

Prove that (4, 3), (6, 4), (5, 6) and (3, 5) are the angular points of a square.

view answer >
17

Prove that the points (-4, -1), (-2, -4), (4, 0) and (2, 3) are the vertices of a rectangle.

view answer >
18

Find the lengths of the medians of a triangle whose vertices are A (-1,3), B (1,-1) and C(5,1).

view answer >
19

Three vertices of a parallelogram are (a + b, a - b), (2a + b, 2a - b), (a - b, a + b). Find the fourth vertex.

view answer >
20

If two vertices of a parallelogram are (3, 2), (-1, 0) and the diagonals cut at (2, -5), find the other vertices of the parallelogram.

view answer >
21

If the coordinates of the mid-points of the sides of a triangle are (3, 4), (4, 6) and (5, 7), find its vertices.

view answer >
22

The line segment joining the points P (3, 3) and Q (6, - 6) is trisected at the points A and B such that A is nearer to P. If A also lies on the line given by 2x + y + k = 0, find the value of k.

view answer >
23

If the points (-2 , -1), (1, 0), (x, 3) and (1, y) form a parallelogram, find the values of x and y.

view answer >
24

The points A (2, 0), B (9, 1), C (11, 6) and D (4, 4) are the vertices of a quadrilateral ABCD. Determine whether ABCD is a rhombus or not.

view answer >
25

If three consecutive vertices of a parallelogram are (1, -2), (3, 6) and (5, 10), find its fourth vertex.

view answer >
26

If the points A(a, -11), B(5, b), C(2, 15) and D(1, 1) are the vertices of a parallelogram ABCD, find the values of a and b.

view answer >
27

If the coordinates of the mid-points of the sides of a triangle be (3, -2), (-3, 1) and (4, -3), then find the coordinates of its vertices.

view answer >
28

Find the lengths of the medians of a ABC having vertices at A (0,-1), B (2, 1) and C (0, 3).

view answer >
29

Find the lengths of the medians of a ABC having vertices at A (5, 1), B (1, 5), and C(-3, -1).

view answer >
30

Find the coordinates of the points which divide the line segment joining the points (-4, 0) and (0, 6) in four equal parts.

view answer >
31

Show that the mid-point of the line segment joining the points (5, 7) and (3, 9) is also the mid-point of the line segment joining the points (8, 6) and (0, 10).

view answer >
32

Find the distance of the point (1, 2) from the mid-point of the line segment joining the points (6, 8) and (2, 4).

view answer >
33

If A and B are (1, 4) and (5, 2) respectively, find the coordinates of P when AP/BP = 3/4.

view answer >
34

Show that the points A (1, 0), B (5, 3), C (2, 7) and D (-2, 4) are the vertices of a parallelogram.

view answer >
35

Determine the ratio in which the point P (m, 6) divides the join of A(-4, 3) and B(2, 8). Also, find the value of m.

view answer >
36

Determine the ratio in which the point (-6, a) divides the join of A(-3, 1) and B(-8, 9). Also find the value of a.

view answer >
37

The line segment joining the points (3, -4) and (1, 2) is trisected at the points P and Q . If the coordinates of P and Q are (p, -2) and (5/3, q) respectively. Find the values of p and q.

view answer >
38

The line joining the points (2,1) and (5,-8) is trisected at the points P and Q. If point P lies on the line 2x - y + k = 0. Find the value of k.

view answer >
39

If A and B are two points having coordinates (-2, -2) and (2, -4) respectively, find the coordinates of P such that AP = AB.

view answer >
40

Find the coordinates of the points which divide the line segment joining A (-2, 2) and B (2, 8) into four equal parts.

view answer >
41

A (4, 2), B (6, 5) and C (1, 4) are the vertices of ABC.

(i) The median from A meets BC in D. Find the coordinates of the point D.

(ii) Find the coordinates of point P on AD such that AP : PD = 2 :1.

(iii) Find the coordinates of the points Q and R on medians BE and CF respectively such that BQ : QE = 2 : 1 and CR : RF = 2 : 1.

(iv) What do you observe?

view answer >
42

ABCD is a rectangle formed by joining the points A (-1, -1), B (-1, 4), C (5, 4) and D (5,-1). P, Q, R and S are the mid-points of sides AB, BC, CD and DA respectively. Is the quadrilateral PQRS a square? a rectangle? or a rhombus? Justify your answer.

view answer >
43

Show that A(-3, 2), B (-5, -5), C (2, -3) and D(4, 4) are the vertices of a rhombus.

view answer >
44

Find the ratio in which the y-axis divides the line segment joining the points (5, -6) and (-1, -4). Also, find the coordinates of the point of division.

view answer >
45

If the points A (6, 1), B (8, 2), C (9, 4) and D (k, p) are the vertices of a parallelogram taken in order, then find the values of k and p.

view answer >
46

In what ratio does the point (-4, 6) divide the line segment joining the points A (-6, 10) and B(3, -8)?

view answer >
47

Find the coordinates of a point A, where AB is a diameter of the circle whose centre is (2, -3) and B is (1, 4).

view answer >
48

A point P divides the line segment joining the points A (3, -5) and B (-4, 8) such that . If P lies on the line x + y = 0, then find the value of k.

view answer >
49

Find the ratio in which the point P(-1, y) line segment joining A ( -3,10) and B(6, -8) divides it. Also find the value of y.

view answer >
50

Points p, Q, R and S divide the segment joining the points A (1, 2) and B (6, 7) in 5 equal parts. Find the coordinates of the points P, Q and R.

view answer >
51

The mid-point P of the line segment joining the points A (- 10, 4) and B (- 2, 0) lies on the line segment joining the points C (- 9,-4) and D (- 4, y). Find the ratio in which P divides CD. Also, find the value of y.

view answer >
52

Find the ratio in which the point P (x, 2) divides the line segment joining the points A (12,5) and B (4, -3). Also, find the value of x.

view answer >
53

Find the ratio in which the line segment joining the points A (3,-3) and B (-2, 7) is divided by x-axis. Also, find the coordinates of the point of division.

view answer >
54

Find the ratio in which the points P (3/4, 5/12) divides the line segments joining the points A(1/2 , 3/2) and B(2, -5).

view answer >
55

If the points P, Q(x, 7), R, S(6, y) in this order divide the line segment joining A(2, p) and B (7, 10) in 5 equal parts, find x, y and p.

view answer >
1

Find the centroid of the triangle whose vertices are:

(i) (1, 4), (-1, -1), (3, -2)

(ii) (- 2, 3), (2, -1), (4, 0)

view answer >
2

Two vertices of a triangle are (1, 2), (3, 5) and its centroid is at the origin. Find the coordinates of the third vertex.

view answer >
3

Prove analytically that the line segment joining the middle points of two sides of a triangle is equal to half of the third side.

view answer >
4

Prove that the lines joining the middle points of the opposite sides of a quadrilateral and the join of the middle points of its diagonals meet in a point and bisect one another.

view answer >
5

If G be the centroid of a triangle ABC and P be any other point in the plane, prove that PA^{2} + PB^{2} + PC^{2} = GA^{2} + GB^{2} + GC^{2} + 3 GP^{2}.

6

If G be the centroid of a triangle ABC, prove that:

AB^{2} + BC^{2} + CA^{2} = 3 (GA^{2} + GB^{2} + GC^{2})

7

If (-2, 3), (4, -3) and (4, 5) are the mid-points of the sides of a triangle, find the coordinates of its centroid.

view answer >
8

In Fig. 14.40, a right triangle BOA is given. C is the mid-point of the hypotenuse AB. Show that it is equidistant from the vertices 0, A and B.

view answer >

9

Find the third vertex of a triangle, if two of its vertices are at (-3, 1) and (0, -2) and the centroid is at the origin.

view answer >
10

A (3, 2) and B (-2, 1) are two vertices of a triangle ABC whose centroid G has the coordinates (5/3, - 1/3). Find the coordinates of the third vertex C of the triangle.

view answer >
1

Find the area of a triangle whose vertices are

(i) (6, 3), (-3, 5) and (4, - 2)

(ii)

(iii) (a, c + a), (a, c) and (-a, c - a)

view answer >
2

Find the area of the quadrilaterals, the coordinates of whose vertices are

(i) (-3, 2), (5, 4), (7, - 6) and (-5, - 4)

(ii) (1, 2), (6, 2), (5, 3) and (3, 4)

(iii) (-4, - 2), (-3, - 5), (3, - 2), (2, 3)

view answer >
3

The four vertices of a quadrilateral are (1, 2), (-5, 6), (7, -4) and (k, -2) taken in order. If the area of the quadrilateral is zero, find the value of k.

view answer >
4

The vertices of ∆ ABC are (-2, 1), (5, 4) and (2, -3) respectively. Find the area of the triangle and the length of the altitude through A.

view answer >
5

Show that the following sets of points are collinear.

(a) (2, 5), (4, 6) and (8, 8)

(b) (1, -1), (2, 1) and (4, 5).

view answer >
6

Prove that the points (a, 0), (0, b) and (1, 1) are collinear if,

view answer >
7

The point A divides the join of P (-5, 1) and Q (3, 5) in the ratio k : 1. Find the two values of k for which the area of a ABC where B is (1, 5) and C (7, -2) is equal to 2 units.

view answer >
8

The area of a triangle is 5. Two of its vertices are (2, 1) and (3, -2). The third vertex lies on y = x + 3. Find the third vertex.

view answer >
9

If , prove that the points (a, a^{2}), (b,b^{2}),(c, c^{2}) can never be collinear.

10

Four points A (6, 3), B (-3, 5), C (4, - 2) and D (x, 3x) are given in such a way that , find x.

view answer >
11

For what value of a the point (a, 1),(1, -1) and(11, 4) are collinear?

view answer >
12

Prove that the points (a, b),(a_{1},b_{1}) and (a - a_{1}, b - b_{1}) are collinear if ab_{1} = a_{1}b

13

If three points (x_{1}, y_{1}), (x_{2}, y_{2}), (x_{3}, y_{3}) lie on the same line, prove that

14

If (x, y) be on the line joining the two points (1, -3) and (-4, 2), prove that x+y+2=0.

view answer >
15

Find the value of k if points (k, 3), (6, - 2) and (-3, 4) are collinear.

view answer >
16

Find the value of k, if the points A (7, -2), B (5, 1) and C (3, 2k) are collinear.

view answer >
17

If the point P (m, 3) lies on the line segment joining the points A( and B (2, 8), find the value of m.

view answer >
18

If R (x, y) is a point on the line segment joining the points P (a, b) and Q (b, a), then prove that x + y = a + b.

view answer >
19

Find the value of k, if the points A (8, 1), B (3, - 4) and C (2, k) are collinear.

view answer >
20

Find the value of a for which the area of the triangle formed by the points A (a, 2a), B (-2, 6) and C (3, 1) is 10 square units.

view answer >
21

If the vertices of a triangle are (1,-3), (4, p) and (-9, 7) and its area is 15 sq. units, find the value(s) of p.

view answer >
22

Find the area of a parallelogram ABCD if three of its vertices are A(2, 4), B (2 + , 5) and C(2, 6).

view answer >
23

Find the value (s) of k for which the points (3k - 1, k - 2), (k, k - 7) and (k - 1,-k - 2) are collinear.

view answer >
24

If the points A (-1,-4), B (b,c) and C (5,-1) are collinear and 2b + c = 4, find the values of b and c.

view answer >
25

If the points A (-2,1), B (a, b) and C (4,-1) are collinear and a – b = 1, find the values of a and b.

view answer >
26

If A (-3, 5), B (-2,-7), C (1,-8) and D (6, 3) are the vertices of a quadrilateral ABCD, find its area.

view answer >
27

If P (-5, - 3), Q (-4, -6), R (2,-3) and S (1, 2) are the vertices of a quadrilateral PQRS, find its area.

view answer >
28

Find the area of the triangle PQR with Q (3, 2) and the mid-points of the sides through Q being (2, -1) and (1, 2).

view answer >