Geometrical Figures, Chessboard and Grid Questions with Solutions — Set 1 (Q1-Q10)

This set contains Geometrical Figures, Chessboard and Grid Questions with Solutions — Set 1 (Q1-Q10) covering a mix of question types and difficulty levels — from basic to advanced — exactly as asked in real competitive exams.

Solutions are written in a simple, step-by-step notebook style for easy self-study and quick understanding. Each solution is broken down step by step so even the toughest question feels easy. These questions are hand-picked for students preparing for SSC CGL, SSC CHSL, CAT, Bank PO, Bank Clerk, UPSC CSAT, Railway RRB, AMCAT, eLitmus, TCS NQT and all campus placement aptitude tests. International students preparing for GRE, GMAT, SAT, ACT, MAT and all Numerical Reasoning Tests will find these equally useful.

✏️ Attempt each question on your own first — then check the solution below.

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.

1. How many distinct lines can be drawn through 10 points, no three of which are collinear?
Solution:-
Number of distinct straight lines
= ¹⁰C₂
= 45     Answer

2. How many straight lines can be drawn through 10 points, exactly 3 of which are collinear?
Solution:-
Number of straight lines
= ¹⁰C₂ – ³C₂ + 1
= 45 – 3 + 1
= 43         Answer

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?
Solution:-
¹⁰C₂ – ⁵C₂ – ⁵C₂ + 1 + 1
= 45 – 10 – 10 + 2
= 27         Answer

4. Find the number of diagonals of a polygon with 12 sides.
Solution:-
¹²C₂ – 12
= 66 – 12
= 54    Answer

5. What is the number of diagonals of a polygon with 15 sides?
Solution:-
Number of diagonals = ¹⁵C₂ – 15
= 105 – 15
= 90       Answer

6. How many distinct triangle can be drawn with their with their vertices selected from 10 points, exactly 5 of which are collinear?
Solution:-
10 points could have given us ¹⁰C₃ triangles if no three of them are in a straight line, but as per question 5 of them are in a straight line so we will get ⁵C₃ number of triangle less.
Hence required number of triangle is = ¹⁰C₃ – ⁵C₃
= 120 – 10
= 110      Answer

7. Consider a square along with its 2 diagonal. How many triangle is formed by this system?
Solution:-

total vertices i.e. total points = 5
∴ if no three point would have collinear then number of triangle = ⁵C₃
Since there are 2 sets of three collinear points 
hence required number = ⁵C₃ – 2׳C₃
                                            = 10 – 2
                                            = 8       Answer

8. A polygon has 65 diagonals then find its number of sides.
Solution:-
Let number of sides = n
∴ number of vertices = n
⟹ ⁿC₃ – n = 65
⟹  \(\frac{{n\left( {n – 3} \right)}}{2}\) = 65
⟹ n(n-3) = 2×5×13
n(n-3) = 13(13 – 3)

∴ n = 13           Answer

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?
Solution:-
¹⁰C₃ – ⁵C₃ – ⁵C₃
= 120 – 10 – 10
= 100      Answer

10. What is the number of quadrilaterals that can be formed by joining the vertices of a polygon of n sides.
Solution:-
number of vertices in polygon of n sides = n
Hence required number of triangle = ⁿC₄    Answer

Continue practising with Geometrical Figures, Chessboard and Grid Questions 11 to 20 → Set 2 or revisit the Geometrical Figures, Chessboard and Grid Concept Page to strengthen your formulas and tricks before moving ahead.

Consistent practice is the key to mastering Geometrical Figures, Chessboard and Grid for SSC CGL, SSC CHSL, CAT, Bank PO, Bank Clerk, UPSC CSAT, Railway RRB, AMCAT, eLitmus, TCS NQT and international exams including GRE, GMAT, SAT, ACT, MAT and all Numerical Reasoning Tests. Want to understand the concept better? Read about Shortest Path Problem on Wikipedia before attempting the next set.

This page is part of our complete series of Geometrical Figures, Chessboard and Grid questions with solutions for competitive exams — covering every question type from basic to advanced so you can build speed, accuracy and confidence. Practising these questions regularly will also strengthen your core Geometrical Figures, Chessboard and Grid concept before your exam day.