exercise on geometrical construction

1. How many distinct lines can be drawn through 10 points, no three of which are collinear?

2. How many straight lines can be drawn through 10 points, exactly 3 of which are collinear?

3. How many distinct lines can be drawn through 10 points, 5 of which are on a straight line & remaining 5 are on other straight line which is parallel to the first straight line?

4. Find the number of diagonals of a polygon with 12 sides.

5. What is the number of diagonals of a polygon with 15 sides?

6. How many distinct triangle can be drawn with their with their vertices selected from 10 points, exactly 5 of which are collinear?

7. Consider a square along with its 2 diagonals, how many triangle is formed by the system?

8. A polygon has 65 diagonals then find its number of sides.

9. How many distinct triangle can be drawn with their vertices selected from 10 points, exactly 5 of which are on one line and remaining 5 are on other parallel line?

10. What is the number of quadrilaterals that can be formed by joining the vertices of a polygon of n sides.

11. consider a set of two octagons, what is the maximum number of quadrilaterals that can be drawn from these 16 points ( vertices of two octagons ) as vertices such that at least one point is selected from each octagon.

12. There are two parallel lines. First line has two points  A & B while second line has 10 points. How many triangle can be formed taking these 12 points as vertices of triangle?

13. Consider a polygon of ‘n’ sides. If the number of diagonal of polygon is 65 then how many triangles can be drawn from taking vertices of polygon as vertices of triangle?

14. Consider a set of 8 non-overlapping triange in a plane such that no three points in the plane is collinear then
(i). if all the possible triangle are drawn taking vertices of these triangles such that not nore than 1 point is selected from a triangle then find the total number triangle hence drawn?

(ii). Find the total number of new triangles that can be drawn from this system of triangles?

15. Consider a square with 5 points on each side (no point on vertices)
(i). How many straight lines can be drawn from these 20 points such that each line passes through exactly 2 of the given points?
(ii). How many triangle can be drawn from these 20 points as vertices?
(iii). How many quadrilateral can be drawn from these 20 points as vertices?

16. Consider vertices of an octagon & mid point of its sides (total 16 points). How many triangles can be drawn taking these 16 points as vertices of the triangle?

17. If number of triangles that can be drawn from the given set of points as vertices is 56 then find the number of straight lines that can be drawn from these points.

18. Consider a set of ‘m’ parallel lines and another set of ‘n’ parallel lines then what is the number of vertices of parallelogram thus formed by these parallel lines?

19. Consider three set of parallel lines, having a, b, c points respectively & no three apart from the given three are collinear
(i). if p represents number of straight lines that can pass through this system of a + b + c points excluding the three original lines then how many of the following represents correct value of p.
    (1) ᵃ⁺ᵇ⁺ᶜC₂ – (ᵃC₂ + ᵇC₂ + ᶜC₂)
    (2) ab + bc + ca
    (3) ᵃC₁×ᵇC₁ + ᵇC₁×ᶜC1 + ᶜC1×ᵃC₁
    (4) 2(ab + bc + ca)
(ii) what is the number of triangle that can be formed from the system of a + b + c points?
(iii) what is the number of  quadrilaterals that can be formed from the system of a + b + c points?

20. Consider a set of ‘m’ parallel lines and another set of ‘n’ parallel lines then what is the number of parallelograms thus formed by these parallel lines?

21. There are 16 points in a plane out of which 7 of them are collinear. Find the number of quadrilaterals formed by joining these points as vertices?

22. A parallelogram is cut by two set of parallel lines parallel to sides of parallelogram. Each set has ‘n’ parallel lines. What is the total number of parallelogram thus formed?

23. Consider n points in a plane no three of which are collinear and the ratio of number of hexagon & octagon that can be formed from these n points is 7:69 then find the value of n.

24. The sides AB, BC, CA of a triangle ABC have 4, 5 and 6 interior points respectively on them. Find the number of triangles that can be constructed using given interior points as vertices?

25. Consider three set of parallel lines in a plane containing ‘x’, ‘y’, ‘z’ parallel lines respectively. What is the highest number of parallelograms that can be formed with these set of parallel lines?

26. Find the number of non-congruent rectangles on a chess board.

27. Consider ‘n’ points in a plane, if number of heptagon is equal to number of octagons then find the number of triangles that can be drawn from these n points.

28. Let Kₙ denotes the number of triangles which can b formedusing the vertices of a regular polygon of ‘n’ sides. Which one of the following could be the value of Pₙ₊₁ – Pₙ₋₁?
(a) 194                      (b) 195
(c)196                       (d) 197

29. Find the number of rectangles in a chess board of 11×11 grid instead of 8×8 grid.

30. Find the number of rectangles in a chess board of 12×14 grid instead of 8×8 grid.

31. Find the number of squares in a chess board of 10×12 Grid instead of 8×8 Grid.

32. In how many ways 2 rooks can be placed on a chessboard such that they are not in attacking position?

33. Consider 37 points in a plane such that no three of them are collinear. A student is told to 1st draw all the possible polyg0ns of same number of sides & then draw all the possible diagonals. What is the maximum number of such diagonals possible? ( if a particular line is diagonal of two polygon then count both of them)

34. Consider a polygon of k sides with n points on each side (no point on the vertices)
(i) How many straight lines can be drawn from these ‘kn’ points such that each line passes through exactly 2 of the given points.
(ii) if ‘T’ is the maximum number of triangles that can be drawn from these kn points as vertices then find the value of ‘T’.
(iii) if ‘Q’ is the maximum number of quadrilaterals that can be drawn from these ‘kn’ points as vertices then find the value of Q.

35. Consider a polygon of n sides. what is the number of triangles that can be drawn taking vertices of these polygons as vertices of triangles and no side of triangles is common with any side of the polygon?

36. Consider ‘n’ straight lines in a plane such that no two of which are parallel and no three of which pass through the same point. How many new straight lines can be drawn from the point of intersection of these straight lines?

37. 150 circles are drawn on a plane. What is the maximum number of regions made by this system of 150 circles

38. The sides AB, BC & CA of a triangle ABC have 3, 4 and 5 interior points respectively on them.The number of triangles that can be constructed using these interior points as vertices will be?

39. A rectangle with sides 2n-1 and 2m-1 is divided into squares of unit length by drawing parallel lines as shown in the diagram, then the number of rectangles possible with odd side length is?

40. Let Tₙ be the number of all possible triangles formed by joining vertices of an n – sided regular polygon. If Tₙ₊₁ – Tₙ = 10, then what will be the value of n.

41. In how many ways 2 + and 2 – signs are filled into 4×4 cell where each cell can contain maximum one character such that each row & column can not contain same sign.

42. There are ‘n’ points in a plane. No three of which are on same straight line. All the possible straight lines are made by joining these ‘n’ points.
(i) What is the maximum number of point of intersection of these straight lines?
(ii) Taking point of intersection of these straight lines as vertices of triangles then what is the maximum number of triangles that can be formed?

43. Consider a 6×6 square which is dissected into 9 rectangles by lines parallel to its sides such that all the rectangles have integer sides. Out of 9 rectangles what is the maximum number of congruent rectangle ?

44. Consider a decomposition of an 8×8 chessboard into p non overlapping rectangles with the following condition:-
Condition (i):- Number of white and number of black squares are same
Condition (ii):-  If aᵢ is the number of white squares in the iᵗʰ rectangle then
a₁ < a₂ < a₃ < ………<aₚ .
(i). Find the maximum possible value of p.
(ii). How many such different cases are possible if p is maximum. 

45. In how many ways two kings one black & one white can be placed on a 8×8 chess board such that they are not on adjacent squares?

46. In how many ways two identical kings can be placed on a 8×8 chess board such that they are not on adjacent squares?

47. In how many ways two queens can be placed on a 8×8 chess board such that they are not able to attack each other (Queens can attack in the same row/column/diagonal)?
(i). one queen is black & other queen is white
(ii). both queens are identical.

48. In how many ways can two squares be chosen on a 8×8 chess board such that they have only one corner common?
49. How many regular polygons can be formed by joining the vertices of a 36 sided regular polygon?