**1. Find the co-ordinates of the mid-point of the line segments joining the following pairs of points:**

**(i) (2,– 3),( – 6,7)**

**(ii) (5,– 11),(4,3)**

**(iii) (a + 3,5b),(2a – 1,3b + 4)**

**Solution:**

_{1},y

_{1}) and (x

_{2},y

_{2}) = {(x

_{1}+x

_{2})/2 ,(y

_{1}+y

_{2})/2}

_{1},y

_{1}) and (x

_{2},y

_{2}) = {(x

_{1}+x

_{2})/2 ,(y

_{1}+y

_{2})/2}

_{1},y

_{1}) and (x

_{2},y

_{2}) = {(x

_{1}+x

_{2})/2 ,(y

_{1}+y

_{2})/2}

**2. The co-ordinates of two points A and B are ( – 3,3) and (12,– 7) respectively. P is a point on the line segment AB such that AP∶ PB = 2∶ 3. Find the co-ordinates of P.**

**Solution:**

_{1}= -3 ,y

_{1}= 3 , x

_{2}= 12 ,y

_{2}= -7 ,m = 2 and n = 3

_{2}+nx

_{1})/(m+n)

_{2}+ny

_{1})/(m+n)

**3. P divides the distance between A ( – 2,1) and B (1,4) in the ratio of 2∶ 1. Calculate the co-ordinates of the point P.**

**Solution:**

_{1}= -2 ,y

_{1}= 1 ,x

_{2 }= 1 ,y

_{2}= 4 ,m = 2 and n = 1

_{2}+nx

_{1})/(m+n)

_{2}+ny

_{1})/(m+n)

**4. (i) Find the co-ordinates of the points of trisection of the line segment joining the point (3,– 3)**

**and (6,9).**

**(ii) 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.**

**Solution:**

_{1}= 3, y

_{1}= -3, x

_{2}= 6, y

_{2}= 9

_{2}+nx

_{1})/(m+n)

_{2}+ny

_{1})/(m+n)

_{2}+ny

_{1})/(m+n)

_{1}= 3,y

_{1}= -4, x

_{2}= 1, y

_{2}= 2

_{2}+nx

_{1})/(m+n)

_{2}+ny

_{1})/(m+n)

**5. (i) The line segment joining the points A (3,2) and B (5,1) is divided at the point P in the ratio 1∶ 2 and it lies on the line 3x – 18y + k = 0. Find the value of k.**

**(ii) A point P divides the line segment joining the points A (3,– 5) and B ( – 4,8) such that AP/PB = k/1 If P lies on the line x + y = 0, then find the value of k.**

**Solution:**

_{1}= 3 ,y

_{1}= 2 , x

_{2}= 5 , y

_{2}= 1 ,m = 1 and n = 2

_{2}+nx

_{1})/(m+n)

_{2}+ny

_{1})/(m+n)

_{1}= 3 , y

_{1}= -5 , x

_{2}= -4 ,y

_{2}= 8 ,m = k and n = 1

_{2}+nx

_{1})/(m+n)

_{2}+ny

_{1})/(m+n)

**6. Find the coordinates of the point which is three-fourth of the way from A (3,1) to B ( – 2,5).**

**Solution:**

_{1}= 3 ,y

_{1 }= 1 , x

_{2}= -2,y

_{2}= 5

_{2}+nx

_{1})/(m+n)

_{2}+ny

_{1})/(m+n)

**7. Point P (3,– 5) is reflected to P’ in the x- axis. Also P on reflection in the y-axis is mapped as P”.**

**(i) Find the co-ordinates of P’ and P”.**

**(ii) Compute the distance P’ P”.**

**(iii) Find the middle point of the line segment P’ P”.**

**(iv) On which co-ordinate axis does the middle point of the line segment P P” lie ?**

**Solution:**

_{1},y

_{1}) and P’’(x

_{2},y

_{2}) be the given points

_{1}+x

_{2})/2

_{1}+y

_{2})/2

**8. Use graph paper for this question. Take 1 cm = 1 unit on both axes. Plot the points A(3,0) and B(0,4).**

**(i) Write down the co-ordinates of A1, the reflection of A in the y-axis.**

**(ii) Write down the co-ordinates of B1, the reflection of B in the x-axis.**

**(iii) Assign the special name to the quadrilateral ABA1B1.**

**(iv) If C is the midpoint is AB. Write down the co-ordinates of the point C1, the reflection of C in the origin.**

**(v) Assign the special name to quadrilateral ABC1B1.**

**Solution:**

**9. The line segment joining A ( – 3,1) and B (5,– 4) is a diameter of a circle whose centre is C. find the co-ordinates of the point C.(1990)**

**Solution:**

_{1}= -3,y

_{1}= 1 ,x

_{2}= 5,y

_{2}= -4

_{1}+x

_{2})/2

_{1}+y

_{2})/2

**10. The mid-point of the line segment joining the points (3m,6) and ( – 4,3n) is (1,2m – 1). Find the values of m and n.**

**Solution:**

_{1}= 3m, y

_{1}= 6 , x

_{2}= -4,y

_{2 }= 3n

_{1}+x

_{2})/2

_{1}+y

_{2})/2

**11. The co-ordinates of the mid-point of the line segment PQ are (1,– 2). The co-ordinates of P are ( – 3,2). Find the co-ordinates of Q.**

**Solution:**

_{2},y

_{2}).

_{1}= -3,y

_{1}= 2 ,x = 1 ,y = -2

_{1}+x

_{2})/2

_{2})/2

_{2}= 2+3 = 5

_{1}+y

_{2})/2

_{2})/2

_{2}

_{2}= -4-2

_{2}= -6

_{2}= -6 are not roots of the equation.

**12. AB is a diameter of a circle with centre C ( – 2,5). If point A is (3,– 7). Find:**

**(i) the length of radius AC.**

**(ii) the coordinates of B.**

**Solution:**

**14. The line segment joining A(-1,5/3) the points B (a,5) is divided in the ratio 1∶ 3 at P, the point where the line segment AB intersects y-axis. Calculate**

**(i) the value of a**

**(ii) the co-ordinates of P.**

**Solution:**

_{1 }= -1 ,y

_{1}= 5/3 ,x

_{2}= a,y

_{2}= 5

_{2}+nx

_{1})/(m+n)

_{2}+ny

_{1})/(m+n)

**15. The point P ( – 4,1) divides the line segment joining the points A (2,– 2) and B in the ratio of 3∶ 5. Find the point B.**

**Solution:**

_{2},y

_{2}).

_{1 }= 2,y

_{1}= -2,x = -4,y = 1

_{2}+nx

_{1})/(m+n)

_{2}+5×2)/(3+5)

_{2}+10)/8

_{2}+10

_{2}= -32-10 = -42

_{2}= -42/3 = -14

_{2}+ny

_{1})/(m+n)

_{2}+5×-2)/(3+5)

_{2}-10)/8

_{2}-10

**16. (i) In what ratio does the point (5,4) divide the line segment joining the points (2,1) and (7 ,6) ?**

**(ii) In what ratio does the point ( – 4,b) divide the line segment joining the points P (2,– 2),Q ( – 14,6) ? Hence find the value of b.**

**Solution:**

_{1}= 2 ,y

_{1}= 1 ,x

_{2}= 7,y

_{2}= 6,x = 5,y = 4

_{2}+nx

_{1})/(m+n)

_{1 }= 2 ,y

_{1}= -2 ,x

_{2}= -14,y

_{2}= 6,x = -4,y = b

_{2}+nx

_{1})/(m+n)

_{2}+ny

_{1})/(m+n)

**17. 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 coordinates of the point K.**

**Solution:**

_{1}= 2 ,y

_{1}= 3 ,x

_{2}= 6,y

_{2}= -5,y = 0

_{2}+ny

_{1})/(m+n)

_{2}+ny

_{1})/(m+n)

_{1}= 3,y

_{1}= -1

_{2}= 8 y

_{2}= 9

_{2}+nx

_{1})/(m+n)

_{2}+ny

_{1})/(m+n)

**20. Given a line segment AB joining the points A ( – 4,6) and B (8,– 3). Find:**

**(i) the ratio in which AB is divided by the y-axis.**

**(ii) find the coordinates of the point of intersection.**

**(iii) the length of AB.**

**Solution:**

_{1}= -4,y

_{1}= 6

_{2}= 8,y

_{2}= -3

_{2}+nx

_{1})/(m+n)

**21. (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 1∶ 2.**

**(ii)Calculate the distance OP where O is the origin.**

**(iii) In what ratio does the y-axis divide the line AB ?**

**Solution:**

_{1}= -4,y_1 = 1

_{2}= 17,y

_{2}= 10

_{2}+nx

_{1})/(m+n)

_{2}+ny

_{1})/(m+n)

**22. Calculate the length of the median through the vertex A of the triangle ABC with vertices A (7,– 3),B (5,3) and C (3,– 1).**

**Solution:**

_{1}= 5, y

_{1}= 3

_{2}= 3, y

_{2}= -1

_{1}+x

_{2})/2

_{1}+y

_{2})/2

**23. Three consecutive vertices of a parallelogram ABCD are A (1,2),B (1,0) and C (4,0). Find the fourth vertex D.**

**Solution:**

_{1}= 1, y

_{1}= 2

_{2}= 4, y

_{2}= 0

_{2}+nx

_{1})/(m+n)

_{2}+ny

_{1})/(m+n)

_{1}= 1, y

_{1}= 0

_{1}+x

_{2})/2

_{2})/2

_{2}

_{2 }= 5-1 = 4

_{1}+y

_{2})/2

_{2})/2

_{2}/2

_{2}= 2

_{2}= 2 are not roots of the equation.

**24. If the points A ( – 2,– 1),B (1,0),C (p,3) and D (1,q) form a parallelogram ABCD, find the values of p and q.**

**Solution:**

_{2},y

_{2})

_{1}= 3,y_1 = 2

_{1}+x

_{2})/2

_{2})/2

_{2}= 4

_{2}= 4-3 = 1

_{1}+x

_{2})/2

_{2})/2

_{2}

_{2}= -10-2 = -12

_{2},y

_{2})

_{1}= -1,y_1 = 0

_{1}+x

_{2})/2

_{2}= 4+1 = 5

_{1}+x

_{2})/2

_{2}= -10

_{2}= -10 are not roots of the equation.

**26. 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.**

**Solution:**

**27. Find the third vertex of a triangle if its two vertices are ( – 1,4) and (5,2) and midpoint of one sides is (0,3).**

**Solution:**

**28. Find the coordinates of the vertices of the triangle the middle points of whose sides are (0,½ ) ,( ½ ,½) and ( ½ ,0).**

**Solution:**

_{1},y

_{1}),B(x

_{2},y

_{2}) and C(x

_{3},y

_{3}) be the vertices of the triangle ABC.

_{1}+x

_{2})/2 = 0

_{1}+x

_{2}= 0

_{1}= -x

_{2}..(i)

_{1}+y

_{2})/2 = ½

_{1}+y

_{2}= 1 …(ii)

_{1}+x

_{3})/2 = ½

_{1}+x

_{3}= 1 …(iii)

_{1}+y

_{3})/2 = 0

_{1}+y

_{3}= 0

_{1}= -y

_{3}…(iv)

_{2}+x

_{3})/2 = ½

_{2}+x

_{3}= 1 …(v)

_{2}+y

_{3})/2 = ½

_{2}+y

_{3}= 1 …(vi)

_{2}+x

_{3}= 1

_{2}+x

_{3}= 1

_{3}= 2

_{3}= 2/2 = 1

_{3}= 1 in (iii), we get x

_{1}= 0

_{2}= 0 [From (i)]

_{1}= 0,x

_{2}= 0,x

_{3}= 1

_{3}+y

_{2}= 1

_{2}+y

_{3}= 1

_{2}= 2

_{2}= 2/2 = 1

_{2}= 1 in (i), we get y

_{1}= 0

_{3}= 0

_{1}= 0,y

_{2}= 1,y

_{3}= 0

**29. Show by section formula that the points (3,– 2),(5,2) and (8,8) are collinear.**

**Solution:**

_{2}+nx

_{1})/(m+n)

_{2}+ny

_{1})/(m+n)

**30. Find the value of p for which the points ( – 5,1),(1,p) and (4,– 2) are collinear.**

**Solution:**

_{2}+nx

_{1})/(m+n)

_{2}+ny

_{1})/(m+n)

**31. A (10,5),B (6,– 3) and C (2,1) are the vertices of triangle ABC. L is the mid point of AB,M is the mid-point of AC. Write down the co-ordinates of L and M. Show that LM = ½ BC.**

**Solution:**

_{1}= 10,y

_{1}= 5

_{2}= 6,y

_{2}= -3

_{1}+x

_{2})/2

_{1}+y

_{2})/2

_{1}= 10,y

_{1}= 5

_{2}= 2,y

_{2}= 1

_{1}+x

_{2})/2

_{1}+y

_{2})/2

**32. A (2,5),B ( – 1,2) and C (5,8) are the vertices of a triangle ABC.P and Q are points on AB and AC respectively such that AP∶ PB = AQ∶ QC = 1∶ 2.**

**(i) Find the co-ordinates of P and Q.**

**(ii) Show that PQ = 1/3 BC**

**Solution:**

_{1}= 2,y

_{1}= 5

_{2}= -1,y

_{2}= 2

_{2}+nx

_{1})/(m+n)

_{2}+ny

_{1})/(m+n)

_{1}= 2,y

_{1}= 5

_{2}= 5,y

_{2}= 8

_{2}+nx

_{1})/(m+n)

_{2}+ny

_{1})/(m+n)

**33. The mid-point of the line segment AB shown in the adjoining diagram is (4,– 3). Write down the co-ordinates of A and B.**

**Solution:**

_{2},0)

_{1})

_{1}+x

_{2})/2

_{2})/2

_{2}= 4×2 = 8

_{1}+y

_{2})/2

_{1}+0)/2

_{1}= -3×2 = -6

_{1}=-6 are not roots of the equation.

**34. Find the co-ordinates of the centroid of a triangle whose vertices are A ( – 1,3),B(1,– 1) and C (5,1)**

**Solution:**

_{1},y

_{1}),(x

_{2},y

_{2}) and (x

_{3},y

_{3}) are

_{1}+ x

_{2}+ x

_{3})/3,(y

_{1}+ y

_{2}+ y

_{3})/3]

_{1},y

_{1}) = (-1,3)

_{2},y

_{2}) = (1,-1)

_{3},y

_{3}) = (5,1)

_{1}+ x

_{2}+ x

_{3})/3 = (-1+1+5)/3 = 5/3

_{1}+ y

_{2}+ y

_{3})/3] = (3-1+1)/3 = 3/3 = 1

**36. The vertices of a triangle are A ( – 5,3),B (p,– 1) and C (6,q). Find the values of p and q if the centroid of the triangle ABC is the point (1,– 1).**

**Solution:**

_{1},y

_{1}),(x

_{2},y

_{2}) and (x

_{3},y

_{3}) are

_{1}+ x

_{2}+ x

_{3})/3,(y

_{1}+ y

_{2}+ y

_{3})/3]

_{1},y

_{1}) = (-5,3)

_{2},y

_{2}) = (p,-1)

_{3},y

_{3}) = (6,q)

_{1}+ x

_{2}+ x

_{3})/3 = (-5+p+6)/3 = 1

_{1}+ y

_{2}+ y

_{3})/3 = (3-1+q)/3 = -1