1. If A = (-4, 3) and B = (8, -6)
   (i) find the length of AB  
  (ii) In what ratio is the line joining AB, divided by the x-axis?

     A = (-4, 3), B = (8, -6)
    
    Join AB. It passes through the origin O  (0, 0)
    Let O divides AB in the ratio m₁ : m₂
    

    
     Hence O, divides AB in the ratio 1 : 2

2. A(-3, 4), B(3, -1) and C(-2, 4) are the vertices of triangle ABC. Find the length of line segment AP, where point P lies inside BC, such that BP : PC = 2 : 3.

     
     Let, the co-ordinates of P be (x, y) which lies inside BC and BP : PC = 2 : 3

3. The line segment joining A(2, 3) and B(6, -5) is intercepted by the x-axis at the point K. Write the ordinate of the point K. Hence find the ratio in which K divides AB. Also, find the co-ordinates of the point K.

     Let the line segment Intersect the x-axis at the point P.
     ∴ Co-ordinates of P are (x, 0)
     Let P divide the line segment in the ratio K : 1 then 

   
    Hence, required ratio is 3 : 5

4. The line segment joining A (4, 7) and B (-6, -2) is intercepted by the y - axis at the point K. Write down the abscissa of the point K. Hence, find the ratio in which K divides AB.
Also, find the co-ordinates of the point K.

    Points A (4, 7), B (-6, -2) are joined which intersects y-axis at K.
    ∴ abscissa of K will be 0

         Let the coordinates of K be (0, y) and K divides AB line segment in the ratio m₁ : m₂
        

5. A(5, 3), B(-1, 1) and C(7, -3) are the vertices of ΔABC. If L is the mid-point of AB and M is the midpoint of AC, show that LM = 

      Vertices of Δ ABC are A (5, 3), B(-1, 1) and C (7, -3)
      (i) When L (x, y) the mid-point of AB
       

      (ii) When M (x, y) is the mid-point of AC
        

6. P(-3,2) is the mid-point of line segment AB as shown in the figure. Find the co-ordinates of points A and B.
                     

                     
    ∵  Point A is on y-axis
   ∴  its abscissa is zero and point B is on x-axis
   ∴  its ordinate is zero.
     Now, let co-ordinates of A are (0, y) and of B are (x, 0) and P (-3, 2) is the mid-point
      
     ∴ Co-ordinates of A are (0, 4) and B(-6, 0)

7. In the given figure, P (4, 2) is the mid-point of line segment AB. Find the co-ordinates of A and B.
      

     
     ∵ Points A and B are on x-axis and y-axis respectively.
     ∴ Ordinate of A is zero and abscissa of B is zero.
     Let co-ordinates of A be (x, 0) and B (0, y) and P (4, 2) is the mid-point
     
     ∴ Co-ordinates of A are (8, 0) and of B are (0, 4).

8. (-5, 2), (3, -6) and (7, 4) are the vertices of a triangle. Find the lengths of all its medians     .

     
     Let A (-5, 2), B(3, -6) and C (7, 4) are the vertices of a ∆ABC
     Let L, M and N are the mid-points of sides BC, CA and AB respectively of ∆ABC.
     ∵ L is the mid-point of BC.
     ∴ Co-ordinates of L will be
   
   ∵ M is the mid-point of CA.
   ∴ Co-ordinates of M will be
  
    ∵ N is the mid-point of AB.
   ∴ Co-ordinates of N will be
     

9. Given a line ABC in which AB = BC = CD, B = (0, 3) and C = (1, 8).
   Find the co-ordinates of A and D.
      

    (i) Let Co-ordinates of A be (x, y)
     ∴ AB = BC or B is the mid-point of AC.
     
     ∴ y = 6 - 8 = -2
     ∴ Co-ordinates of A are (-1, -2)

     (ii) Let co-ordinates of D be (x, y)
       But BC = CD or C is the mid point of BD
         
     ∴ y =16 - 3 = 13
     ∴ Co-ordinates of D are (2, 13).

10. One end of the diameter of a circle is (-2, 5). Find the co-ordinates of the other end of it, if the centre of the circle is (2, -1).

     Let O(2, -1) be the centre and A (-2, 5) and B (x, y) be the two end points of the diameter AB of the circle.
     ∵ O is the mid-point of AB
      
      ∴ Co-ordinates of B are (6, -7)

11. P (4, 2) and Q (-1, 5) are the vertices of parallelogram PQRS and (-3, 2) are the co-ordinates of the point if intersection of its diagonals. Find the co-ordinates of R and S.

      In the parallelogram PQRS and co-ordinates of P are (4, 2) and of Q are (-1, 5). The diagonals of ||gm AC and BD intersect each other at O (-3, 2)

      ∴ O is the mid-point of AC as well as of BD.
      Let, the co-ordinates of R be (x₁, y₁)
     
       ∴ y₁ = 4 - 2 = 2
       ∴ Co-ordinates of R are (-10, 2)
       Again, let the co-ordinates of S be (x₂, y₂)
        
      ∴ y₂ = 4 - 5 = -1
      ∴ Co-ordinates of R are (-5, -1)

12. The points (2, -1), (-1, 4) and (-2, 2) are the mid-points of the sides of a triangle. Find its vertices.

      Let D, E and F are the mid-points of sides BC, CA and AB of a ΔABC respectively.

      Then, co-ordinates of D(2, -1), E(-1, 4) and F (-2, 2).
      Let, co-ordinates of A (x₁, y₁), B(x₂, y₂) and C(x₃, y₃), then
     
       Now adding (i) (iii) and (v) we get
      2(x₁ + x₂ + x₃) = -4 + 4 - 2 = -2
     ∴ x₁ + x₂ + x₃ = -1                                        (vii)
     Subtracting (i), (iii) and (v) turn by turn from (vii)
     x₃ = 3, x₁ = -5, x₂ = 1
      Again adding (ii), (iv) and (vi), we get:
     2(y₁ + y₂ + y₃) = 4 - 2 + 8 = 10
    ∴ y₁ + y₂ + y₃ = 5                                          (viii)
    Now, subtracting (ii), (iv) and (vi) turn bu turn from (viii) we get :
     y₃ = 1, y₁ = 7, y₂ = -3
    ∴ Co-ordinates of A are (-5, 7), of B are (1, -3) and of C are (3, 1).

13. The co-ordinates of the centroid of a triangle PQR are (2, -5). If Q = (-6, 5) and R = (11, 8); calculate the co-ordinates of vertex P.

      Let co-ordinates of P be (x, y) and of centroid G are (2, -5)
      
      Hence, co-ordinates of vertex P are (1, -28)

14. A(5, x), B(-4, 3) and C(y, -2) are the vertices of the triangle ABC whose centroid is the origin. Calculate the values of x and y.

       Co-ordinates of centroid of ΔABC are (0, 0)

        

15. The points A (-1, 2), B(x, y) and C (4, 5) are such that BA = BC. Find the linear relation between x and y.

      A(-1, 2), B (x, y) and C (4, 5)
     
     Squaring on both sides, we get
     (-1 - x)² + (2 - y)² = (4 - x)² + (5 - y)²
       ⟹1 + 2x + x² + 4 - 4y + y² = 16 - 8x + x² + 25 - 10y + y²
      ⟹ 2x - 4y + 5 = -8x - 10y + 41
      ⟹ 2x - 4y + 8x + 10y = 41 - 5 ⟹ 10x + 6y = 36
      ⟹ 5x + 3y = 18

16. Given a triangle ABC in which A = (4, -4), B = (0, 5) and C = (5, 10). A point P lies on BC such that BP : PC = 3 : 2. Find the length of line segment AP.

    B(0, 5), C(5, 10) and BP : PC = 3 : 2
    ∴ Co-ordinates of P will be

17. A line segment joining A and B (a, 5) is divided in the ratio of 1 : 3 at P, the point where the line segment AB intersects the y-axis.
    (i) Calculate the value of 'a'
   (ii) Calculate the co-ordinates of 'P'.

     A , B (a, 5)

       P divides AB in ratio of 1 : 3
       ∵ Co-ordinates of P will be
        

18. In what ratio is the line joining A (0, 3) and B(4, -1), divided by the x-axis? Write the co-ordinates of the point where AB intersects the x-axis.

     Let the ratio be m₁ : m₂ when the x-axis intersects the line AB at P.

       ∴ Let co-ordinate of P (x, 0)

19. The mid point of the segment AB, as shown in diagram, is C (4, -3). Write down the co-ordinates of A and B.

      Let co-ordinates of A (x, 0) and B (0, y) and C (4, -3) the mid point of AB.

20. (i) Write down the co - ordinates of the point P that divides the line joining A(-4, 1) and B(17, 10) in the ratio of 1 : 2.
    (ii) Calculate the distance OP, where O is the origin.  
   (iii) In what distance does the y-axis divide the line AB?

      Point P, divides a line segment giving the points A (-4, 1) and B (17, 10) is the ratio 1 : 2.

        (i) Let co-ordinates of P be (x, y), then
        

      (ii) O (0, 0) is the origin
      

       (iii) Line AB intersects y-axis at L
           ∴ abscissa of L is zero
          Let co-ordinates of L be (0, y) and let L divides AB in the ratio m₁ : m₂
           

21. Prove that the points A(-5, 4); B(-1, -2) and C (5, 2) are the vertices of an isosceles right-angled triangle.Find the co-ordinates of D. So that ABCD is a square.

          In ΔABC, the co-ordinates of A, B and C are (-5, 4), b(-1, -2) and C(5, 2) respectively.

         
      ∵ AB² + BC² = CA²
      ∴ ΔABC is also a right-angled triangle.
        Hence ΔABC is an isosceles right angled triangle,
        Let D be the fourth vertex of square ABCD and co-ordinates of D be (x, y)
        Since the diagonals of a square bisect each other and let O be the point of intersection of AC and BD.
        ∴ O is mid-point of AC as well as BD.