CBSE Class 10 Mathematics Coordinate Geometry Worksheet Set C

CBSE Class 10 Mathematics Worksheet - Coordinate Geometry (3)

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CASE STUDY - 1

Two friends Dalvin and Alice works in the same office in Toronto. In the Christmas vacation, they both decided to go to their home towns represented by Town X and Town Y. Town X and Town Y are connected by trains from the same station C in Toronto. The situation of Town X, Town Y and station A is shown on the coordinate axis.
Based on the given situation, answer the following questions:

Question. What is the distance that Dalvin have to travel to reach his hometown X ?
(a) √51 units
(b) √53 units
(c) √35 units
(d) √47 units
Answer : B

Question. What is the distance that Alice has to travel to reach her hometown Y?
(a) 2√26 units
(b) √107 units
(c) 2√10 units
(d) √51 units
Answer : A

Question. Now, both of them plan to meet at a place between Town X and Town Y, such that it is a mid-point between both. Calculate the coordinates of the mid-point of X and Y.
(a) (1, 3)
(b) (2, - 4)
(c) (2.5,3)
(d) (3.5, 4)
Answer : D

Question. While travelling from A to Y, Alice had to change the train, at a station, it divides the line AY in the ratio of 2: 3, find the position of station on the grid.
(a) (0,79)
(b) ( − 115,245)
(c) (118,173)
(d) (12, 7)
Answer : B

Very Short Answer type Questions

Question. A (5,1), B (1,5) and C (-3, -1) are the vertices of 𝛥ABC. Find the length of median AD.
Answer : √37 units

Question. Find the point on the x-axis which is equidistant from the points (2, -5) and (-2, 9)
Answer : (-7, 0)

Question. Find the distance of the point P (2, 3) from the x-axis.
Answer : 3

Question. Find the perimeter of a triangle with vertices (0,4), (0,0) and (3,0).
Answer : 12

Question. In what ratio does the point P (2, -5) divide the line segment joining A (-3, 5) and B (4, -9).
Answer : k = 5/2 or k = 5 : 2

Question. In a seating arrangement of desks in a classroom, three students are seated at A (3, 1), B (6,4) and C (8, 6) respectively. Are they seated in line?
Answer : Yes

Question. If the point P (x, y) is equidistant from the points A (a + b, b - a) and B (a - b, a + b), then prove that 𝑏𝑥 = 𝑎𝑦.
Answer : bx = ay.

Question. If the mid-point of the line segment joining A ( 𝑥/2, 𝑦 + 1/2 ) and B ( 𝑥 + 1,𝑦 − 3) is C (5, -2), find x, y.
Answer : y = -1

Question. Find the ratio in which the line segment joining the points P (3, -6) and Q (5,3) is divided by the x -axis.
Answer : 2 : 1

Question. Find the perpendicular distance of A (5, 12) from the y -axis.
Answer : 5 units

Short Answer type Questions

Question. The vertices of quadrilateral ABCD are A (5, -1), B (8,3), C (4, 0) and D (1, -4). Prove that ABCD is a rhombus.
Answer : The sides of the quadrilateral AB = BC = CD = AD = 5 units & the
diagonals AC = √2 units and BD = 7√2 units
As the length of all the sides are equal and the length of the diagonals are not equal.
⇒ ABCD is a rhombus

Question. The base QR of an equilateral triangle PQR lies on x-axis. The co-ordinates of point Q are (-4, 0) and the origin is the mid-point of the base. Find the co-ordinates of the point P and R.
Answer : Coordinates of P are (0, 4√3 ) or (0, − 4√3 )

Question. Find the centre and radius of the circumcircle (i.e., circumcentre and circum-radius) of the triangle whose vertices are (-2, 3), (2, -1) and (4, 0).
Answer : Circumcentre of the 𝛥ABC is (3/2 , 5/2) and Circumradius of 𝛥ABC is 5√2/2

Question. The three vertices of a parallelogram ABCD are A (3, -4), B (-1, -3) and C (-6, 2). Find the coordinates of vertex D and find the area of ABCD.
Answer : 15 square units

Question. Find the coordinates of the points of trisection (i.e., Points dividing in three equal parts) of the line segment joining the points A (2, -2) and B (-7, 4).
Answer : The coordinates of the points of trisection of the line segment joining A and B are (-1, 0) and (-4, 2)

Question. An equilateral triangle has one vertex at (3, 4) and another at (-2, 3). Find the co-ordinates of the third vertex.
Answer : Third vertex has the coordinates (1+√3/2, 7 − 5√3/2) or (1 − √3/2, 7 + 5√3/2)

Question. 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 x.
Answer : Ratio is 3 :5 and x = 9

1) Show that the points (a, a), (-a, - a) and (- √3a, √3a) are the vertices of an equilateral Δ

2) Show that four points (0,-1), (6, 7), (-2, 3) and (8, 3) are the vertices of a rectangle

3) Prove that the diagonals of a rectangle with vertices (0, 0), (a, 0), (a, b) and (0, b) bisect each each other and are equal.

4) Prove that (4, -1), (6, 0), (7, 2) and (5, 1) are the vertices of a rhombus. Is it a square?

5 Show that the points A(3 , 5), B( 6 , 0) ,C( 1 ,-3) and D(-2 , 2) are the vertices of a square ABCD

6) Show that the following points are the vertices of a right angled isosceles triangle: (1, 2), (1, 5) and (4, 2)

7) Find a relation between x and y such that the point (x, y) is equidistant from the points (7, 1) and (3, 5) (x - y = 2)

8) If the distance of P(x, y) from the points A (3, 6) and B (-3, 4) are equal, prove that 3x + y = 5

9) Find the values of x for which the distance between the points P (2, -3) and Q (x, 5) is 10 units (8 or -4)

10) Given A (-2, 3) and AB = 10 units .If ordinate of B is 9, find abscissa of B (-10, 6)

11) Find the coordinates of the point equidistant from three given points A (5, 1), B (-3, -7) and C (7, -1) (2,-4) 12) If the point p(x, y) is equidistant from the points A (a + b, b - a) and B (a - b, a + b), prove that b x = a y

13) Find the point on y- axis which is equidistant from the point (5, -2) and (-3, 2) (0, -2)

14) Find the point on x- axis which is equidistant from the points (2, -5) and (-2, 9) (-7, 0)

15) If the points A (4, 3), and B(x, 5) are on the circle with the centre. O (2, 3), find the value of x (x=2)

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

17) If (1,2) (4, y), (x,6) and (3,5) are the vertices of a parallelogram taken in order , find the value of x and y (x =6, y = 3)

18) Find the value of k for which the points (7, -2), (5, 1), and (3, k) are collinear. (k = 4)

19) Find the value of m, for which the points with co-ordinates (3, 5), (m, 6) and [1/2, 15/2] are collinear (m = 2)

20) Find a relation between x and y, if (x, y), (1, 3) and (8, 0) are collinear (3x +7y = 24)

21) If the points (-2, 1), (a, b) and (4,-1) are collinear and a - b = 1, then find the values of a and b (a =1, b = 0)

22) Check whether the points (4, 5), (7, 6) and (6, 3) are collinear.

23) Show that the point P ( - 4, 2 ) lies on the line segment joining the points A( -4, 6) and B( -4 ,-6)

24) If P(x, y) is any point on the line joining the points A (a, 0), B (0, b), then show that x + y

a b

25) If A (-5, 7), B (-4, -5), C (-1, -6) and D (4, 5) are the vertices of a quadrilateral, find the area of the quadrilateral ABCD.

26) ABCDE is polygon whose vertices are A (-1, 0), B (4, 0), C (4, 4), D (0, 7) and E (-6, 2). Find the area of the polygon

27) Using A (4, -6), B (3, -2) and C (5, 2), verify that a median of the ΔABC divides it into two triangles of equal areas

28) The coordinates of A, B, C are (3, 4), (5, 2), (x, y) respectively. If area of ΔABC = 3, show that x + y = 10

29) The coordinates of the vertices of ΔABC are A (4, 1), B (-3, 2) and C (0, k).Given that the area of ΔABC is 12 unit2, Find the

Value of k (k = - 13/ 7)

30) The points A(2,9), B(a,5) , C(5,5) are the vertices of a triangle ABC right angled at B . Find the valueof a and hence the area

of ΔABC (a =2, area = 6sq units)

31) If point P (1/2, y) lies on the line segment joining two points A (3, -2) and B (-7, 9), then find the ratio in which P divides AB.

Also find the value of y

32) Find the ratio in which the point (2, y) divides the line segment joining the points A (-2, 2) and B (3, 7) (4: 1) 33) Find the ratio in which the line 2x + y – 5 = 0 divides the line segment joining A (2,-3) and B (3, 9) (2:5)

34) Determine the ratio in which the line 3x + 4y – 9 = 0 divides the line segment joining the points (1, 3) and (2, 7) (k = - 6/25)

35) Find the ratio in which the line segment joining the points (1, -3) and (4, 5) is divided by x - axis

36) If P divides the join of A (-2, -2) and B (2, -4) such that AP/AB = 3/7, find the coordinates of P (-2/7, -20/7)

37) Find the coordinates of the points which divide the line segment joining A (2, -3) and B (-4, -6) into three equal parts

38) Find the length of medians of triangle whose vertices are A (-1, 3), B (1, -1), and C (5, 1)

39) If the midpoint of of the segment joining A (a, b +1), and B (a +1, b +2) is C (3/2, 5/2) Find a and b (a=1, b=1)

40 The coordinates of one end point of a diameter of a circle are (4, -1) and the coordinates of the centre of the circle are (1, -3)

Find the coordinates of the other end of the diameter (-2, -5)

41) The centre of a circle is (2a – 1, 7) and it passes through the point (-3, -1). If the diameter of the circle is 20 units, then find

the value of a

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