CBSE Class 9 Mathematics

Chapter-wise Important Formulas & Key Points

CHAPTER 1: NUMBER SYSTEMS

Topic	Important Formulas / Key Points
1.1 Introduction	Natural Numbers (N): 1, 2, 3, 4,... Whole Numbers (W): 0, 1, 2, 3,... Integers (Z): ...-3, -2, -1, 0, 1, 2, 3,... Rational Numbers (Q): Numbers in the form $p / q$ , where $p$ and $q$ are integers and $q \neq 0$ .
1.2 Irrational Numbers	Definition: A number $s$ is irrational if it cannot be written in the form $p / q$ . Examples: $\sqrt{2}, \sqrt{3}, π, 0.1010010001...$
1.3 Real Numbers and their Decimal Expansions	Terminating Decimal: Denominator (in simplest form) has prime factors only 2 or 5. Non-terminating Repeating (Rational): e.g., $0. \overline{3}$ Non-terminating Non-repeating (Irrational): e.g., $0.1010010001...$
1.4 Operations on Real Numbers	Operations: Real numbers follow commutative, associative, and distributive laws. Key Fact: The sum or difference of a rational and an irrational number is irrational. The product or quotient of a non-zero rational and an irrational number is irrational.
1.5 Laws of Exponents for Real Numbers	For positive real numbers $a$ and $b$ , and rational numbers $m$ and $n$ : • $a^{m} \times a^{n} = a^{m + n}$ • $a^{m} \div a^{n} = a^{m - n}$ • $(a^{m})^{n} = a^{m n}$ • $a^{m} \times b^{m} = (a b)^{m}$ • $a^{0} = 1$ • $a^{- m} = \frac{1}{a^{m}}$

CHAPTER 2: POLYNOMIALS

Topic	Important Formulas / Key Points
2.1 Introduction	Polynomial: An algebraic expression where the exponents of variables are whole numbers. $p (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + . . . + a_{1} x + a_{0}$
2.2 Polynomials in One Variable	Degree: The highest power of the variable. Constant Polynomial: Degree 0 (e.g., 5) Linear Polynomial: Degree 1 (e.g., $a x + b$ ) Quadratic Polynomial: Degree 2 (e.g., $a x^{2} + b x + c$ ) Cubic Polynomial: Degree 3 (e.g., $a x^{3} + b x^{2} + c x + d$ )
2.3 Zeroes of a Polynomial	A real number $k$ is a zero of $p (x)$ if $p (k) = 0$ . Geometrical Meaning: The x-coordinate of the point where the graph intersects the x-axis.
2.4 Factorisation of Polynomials	Factor Theorem: $x - a$ is a factor of $p (x)$ if and only if $p (a) = 0$ . Remainder Theorem: If $p (x)$ is divided by $(x - a)$ , the remainder is $p (a)$ .
2.5 Algebraic Identities	(a + b)² = a² + 2ab + b² (a – b)² = a² – 2ab + b² a² – b² = (a + b)(a – b) (x + a)(x + b) = x² + (a + b)x + ab (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca (a + b)³ = a³ + b³ + 3ab(a + b) (a – b)³ = a³ – b³ – 3ab(a – b) a³ + b³ = (a + b)(a² – ab + b²) a³ – b³ = (a – b)(a² + ab + b²)

CHAPTER 3: COORDINATE GEOMETRY

Topic	Important Formulas / Key Points
3.1 Introduction	A system to locate the position of a point in a plane using two perpendicular lines (axes).
3.2 Cartesian System	Origin: The point of intersection of the x-axis and y-axis (0, 0). Abscissa: The x-coordinate (distance from y-axis). Ordinate: The y-coordinate (distance from x-axis). Quadrants: • I: (+, +) • II: (–, +) • III: (–, –) • IV: (+, –) Line Parallel to X-axis: Equation $y = a$ (constant). Line Parallel to Y-axis: Equation $x = a$ (constant).

CHAPTER 4: LINEAR EQUATIONS IN TWO VARIABLES

Topic	Important Formulas / Key Points
4.1 Introduction	An equation that can be put in the form $a x + b y + c = 0$ , where $a, b, c$ are real numbers and $a$ and $b$ are not both zero.
4.2 Linear Equations	Standard Form: $a x + b y + c = 0$ Examples: $2 x + 3 y = 5$ , $x - 2 y = 0$
4.3 Solution of a Linear Equation	Infinitely Many Solutions: Every linear equation in two variables has infinitely many solutions. Graphical Representation: A straight line. Every point (x, y) on the line is a solution, and every solution lies on the line.

CHAPTER 5: INTRODUCTION TO EUCLID’S GEOMETRY

Topic	Important Formulas / Key Points
5.1 Introduction	The study of geometry based on definitions, axioms, and postulates given by Euclid.
5.2 Euclid’s Definitions, Axioms and Postulates	Axioms (7): 1. Things equal to the same thing are equal. 2. If equals are added, the wholes are equal. 3. If equals are subtracted, the remainders are equal. 4. Things which coincide are equal. 5. The whole is greater than the part. 6. Things double of the same are equal. 7. Things halves of the same are equal. Postulates (5): 1. A straight line may be drawn from any one point to any other. 2. A terminated line can be produced indefinitely. 3. A circle can be drawn with any centre and any radius. 4. All right angles are equal. 5. If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two lines, if produced indefinitely, meet on that side.

Topic

Important Formulas / Key Points

5.1 Introduction

The study of geometry based on definitions, axioms, and postulates given by Euclid.

5.2 Euclid’s Definitions, Axioms and Postulates

Axioms (7):
1. Things equal to the same thing are equal.
2. If equals are added, the wholes are equal.
3. If equals are subtracted, the remainders are equal.
4. Things which coincide are equal.
5. The whole is greater than the part.
6. Things double of the same are equal.
7. Things halves of the same are equal.

Postulates (5):
1. A straight line may be drawn from any one point to any other.
2. A terminated line can be produced indefinitely.
3. A circle can be drawn with any centre and any radius.
4. All right angles are equal.
5. If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two lines, if produced indefinitely, meet on that side.

CHAPTER 6: LINES AND ANGLES

Topic	Important Formulas / Key Points
6.1 Introduction	Basic geometric figures: point, line, line segment, ray, angle.
6.2 Basic Terms and Definitions	Collinear Points: Points lying on the same line. Angle: Formed by two rays originating from a common endpoint.
6.3 Intersecting Lines and Non-intersecting Lines	Intersecting Lines: Meet at a point. Parallel Lines: Do not meet (distance between them is constant).
6.4 Pairs of Angles	Complementary Angles: Sum = 90° Supplementary Angles: Sum = 180° Linear Pair of Angles: Adjacent angles whose non-common arms are opposite rays. (Sum = 180°) Vertically Opposite Angles: Are equal. Adjacent Angles: Have a common vertex and a common arm.
6.5 Lines Parallel to the Same Line	Theorem: Lines parallel to the same line are parallel to each other.
6.6 Summary	Properties of Parallel Lines cut by a Transversal: • Corresponding angles are equal. • Alternate interior angles are equal. • Alternate exterior angles are equal. • Interior angles on the same side of the transversal are supplementary.

CHAPTER 7: TRIANGLES

Topic	Important Formulas / Key Points
7.1 Introduction	A closed figure with three sides, three angles, and three vertices.
7.2 Congruence of Triangles	CPCT: Corresponding Parts of Congruent Triangles are equal.
7.3 Criteria for Congruence of Triangles	• SAS Congruence: Two sides and the included angle. • ASA Congruence: Two angles and the included side. • AAS Congruence: Two angles and a non-included side. • SSS Congruence: Three sides. • RHS Congruence: Right angle, Hypotenuse, and a side (for right triangles).
7.4 Some Properties of a Triangle	Theorem 7.1 (Angles opposite equal sides): Angles opposite to equal sides of a triangle are equal. Theorem 7.2 (Sides opposite equal angles): Sides opposite to equal angles of a triangle are equal. Inequality Property: In any triangle, the side opposite the larger angle is longer.
7.5 Some More Criteria for Congruence of Triangles	Application of the above criteria in complex proofs. Triangle Angle Sum Theorem: The sum of all interior angles of a triangle is 180°. Exterior Angle Theorem: An exterior angle is equal to the sum of the two opposite interior angles.

CHAPTER 8: QUADRILATERALS

Topic	Important Formulas / Key Points
8.1 Introduction	A quadrilateral is a four-sided polygon. Sum of its angles = 360°.
8.2 Properties of a Parallelogram	Properties: • Opposite sides are equal and parallel. • Opposite angles are equal. • Diagonals bisect each other. Key Theorems: • A diagonal divides a parallelogram into two congruent triangles. • In a parallelogram, opposite sides are equal. • If each pair of opposite sides of a quadrilateral is equal, it is a parallelogram. • In a parallelogram, opposite angles are equal. • If in a quadrilateral, each pair of opposite angles is equal, it is a parallelogram. • The diagonals of a parallelogram bisect each other. • If the diagonals of a quadrilateral bisect each other, it is a parallelogram.
8.3 The Mid-point Theorem	Theorem: The line segment joining the mid-points of two sides of a triangle is parallel to the third side and equal to half of it. Converse: The line drawn through the mid-point of one side of a triangle, parallel to another side, bisects the third side.

CHAPTER 9: CIRCLES

Topic	Important Formulas / Key Points
9.1 Introduction	A circle is a collection of all points in a plane which are at a fixed distance (radius) from a fixed point (centre).
9.2 Angle Subtended by a Chord at a Point	Theorem: Equal chords of a circle subtend equal angles at the centre. Converse: If the angles subtended by two chords at the centre are equal, the chords are equal.
9.3 Perpendicular from the Centre to a Chord	Theorem: The perpendicular from the centre of a circle to a chord bisects the chord. Converse: The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.
9.4 Equal Chords and their Distances from the Centre	Theorem: Equal chords of a circle (or congruent circles) are equidistant from the centre (or centres). Converse: Chords equidistant from the centre of a circle are equal in length.
9.5 Angle Subtended by an Arc of a Circle	Theorem: The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle. Theorem: Angles in the same segment of a circle are equal. Theorem: Angle in a semicircle is a right angle (90°).
9.6 Cyclic Quadrilaterals	Definition: A quadrilateral whose all four vertices lie on a circle. Theorem: The sum of either pair of opposite angles of a cyclic quadrilateral is 180°. Converse: If the sum of a pair of opposite angles of a quadrilateral is 180°, it is cyclic.

CHAPTER 10: HERON’S FORMULA

Topic	Important Formulas / Key Points
10.1 Area of a Triangle – by Heron’s Formula	Semi-perimeter (s): $s = \frac{a + b + c}{2}$ Heron’s Formula: $Area = \sqrt{s (s - a) (s - b) (s - c)}$ Area of an Equilateral Triangle (side = a): $Area = \frac{\sqrt{3}}{4} a^{2}$ Altitude of an Equilateral Triangle: $h = \frac{\sqrt{3}}{2} a$ Altitude of an Isosceles Triangle (equal sides = a, base = b): $h = \sqrt{a^{2} - \frac{b^{2}}{4}}$

CHAPTER 11: SURFACE AREAS AND VOLUMES

Topic	Important Formulas / Key Points
11.1 Surface Area of a Right Circular Cone	Slant Height (l): $l = \sqrt{r^{2} + h^{2}}$ Curved Surface Area (CSA): $π r l$ Total Surface Area (TSA): $π r (l + r)$
11.2 Surface Area of a Sphere	Sphere: • Surface Area = $4 π r^{2}$ Hemisphere: • Curved Surface Area = $2 π r^{2}$ • Total Surface Area = $3 π r^{2}$
11.3 Volume of a Right Circular Cone	Volume: $\frac{1}{3} π r^{2} h$
11.4 Volume of a Sphere	Sphere: • Volume = $\frac{4}{3} π r^{3}$ Hemisphere: • Volume = $\frac{2}{3} π r^{3}$

CHAPTER 12: STATISTICS

Topic	Important Formulas / Key Points
12.1 Graphical Representation of Data	Class Interval: A group into which raw data is condensed. Class Size / Width: Upper Limit – Lower Limit Class Mark (Mid-point): $\frac{Upper Limit + Lower Limit}{2}$ Frequency: The number of observations in a class. Mean: $\overset{ˉ}{x} = \frac{Sum of all observations}{Number of observations} = \frac{\sum x_{i}}{n}$ Median: The middle value when data is arranged in ascending order. • If n is odd: ${(\frac{n + 1}{2})}^{th}$ term. • If n is even: Average of ${(\frac{n}{2})}^{th}$ and ${(\frac{n}{2} + 1)}^{th}$ terms. Mode: The observation that occurs most frequently. Graphical Representations: • Bar Graph • Histogram (for continuous grouped data) • Frequency Polygon (using class marks)

Topic

Important Formulas / Key Points

12.1 Graphical Representation of Data

Class Interval: A group into which raw data is condensed.
Class Size / Width: Upper Limit – Lower Limit
Class Mark (Mid-point):

\frac{Upper Limit + Lower Limit}{2}

Frequency: The number of observations in a class.

Mean:

\overset{ˉ}{x} = \frac{Sum of all observations}{Number of observations} = \frac{\sum x_{i}}{n}

Median: The middle value when data is arranged in ascending order.
• If n is odd:

{(\frac{n + 1}{2})}^{th}

term.
• If n is even: Average of

{(\frac{n}{2})}^{th}

and

{(\frac{n}{2} + 1)}^{th}

terms.

Mode: The observation that occurs most frequently.

Graphical Representations:
• Bar Graph
• Histogram (for continuous grouped data)
• Frequency Polygon (using class marks)

CHAPTER 12: STATISTICS (IMPORTANT ADDITIONS)

12.1 Graphical Representation of Data (Continued)

Topic	Important Formulas / Key Points
Making Data Continuous (Conversion of Class Intervals)	When class intervals are not continuous (i.e., gaps exist between upper limit of one class and lower limit of next class): Formula to find adjustment factor: $Adjustment Factor = \frac{Upper Limit of a Class - Lower Limit of Next Class}{2}$ Steps: 1. Find the difference between the upper limit of a class and the lower limit of the next class. 2. Divide this difference by 2. 3. Subtract this value from all lower limits. 4. Add this value to all upper limits. Example: Given classes: 10-19, 20-29, 30-39 Gap = 20 - 19 = 1 Adjustment = 1 ÷ 2 = 0.5 New Continuous Classes: 9.5-19.5, 19.5-29.5, 29.5-39.5
Adjusted Frequency (When Class Sizes are NOT Equal)	Why needed: To draw a histogram when class widths are different, the heights of bars must be adjusted for fair comparison. Formula: $Adjusted Frequency = \frac{Frequency of the Class}{Class Size (Width) of that Class} \times Minimum Class Size$ Alternate Formula (for proportionality): $Adjusted Frequency = \frac{Frequency}{Class Width}$ (Then draw histogram with adjusted frequency on Y-axis) Example:	Class	Frequency	Class Width	Adjusted Frequency		0-10	5	10	5		10-20	8	10	8		20-40	18	20	$\frac{18}{20} \times 10 = 9$		40-60	12	20	$\frac{12}{20} \times 10 = 6$	Here, minimum class width = 10, so we adjust all frequencies to this width.

Quick Tips for Exams:

For Continuous Intervals: Always check if upper limit of one class = lower limit of next class. If not, apply the adjustment formula before plotting histogram or finding median.
For Histogram with Unequal Classes: You MUST use adjusted frequency on the Y-axis, NOT the original frequency. Otherwise, the graph will be misleading.
Frequency Polygon: Can be drawn:
- Using a histogram (join midpoints of tops of bars)
- Without histogram (plot class marks vs frequency and join points)
Class Mark Formula: $\frac{Lower Limit + Upper Limit}{2}$

Practice • Precision • Performance

CBSE Class 9 Mathematics
Chapter-wise Checklist with Topics (Based on New NCERT & CBSE Exam Pattern)

CHAPTER 1: NUMBER SYSTEMS

✓	Topic	Question / Concept
☐	1.2 Irrational Numbers	Prove that √3 is an irrational number. (Similar to √2, √5)
☐	1.2 Irrational Numbers	Find two rational and two irrational numbers between √2 and √3.
☐	1.3 Real Numbers and their Decimal Expansions	State whether 13/1600 has a terminating decimal expansion.
☐	1.4 Operations on Real Numbers	Simplify: $(3 + \sqrt{3}) (2 + \sqrt{2})^{2}$
☐	1.4 Operations on Real Numbers	Represent √5 on the number line. (Geometrical construction)
☐	1.4 Operations on Real Numbers	Simplify: $3 \sqrt{5} + 2 \sqrt{5} - \sqrt{5}$
☐	1.4 Operations on Real Numbers	Rationalise the denominator: $\frac{1}{\sqrt{3} - \sqrt{2}}$
☐	1.4 Operations on Real Numbers	Find the value of 'a' and 'b': $\frac{5 + 2 \sqrt{3}}{7 + 4 \sqrt{3}} = a - b \sqrt{3}$
☐	1.3 Real Numbers and their Decimal Expansions	If $\frac{1}{7} = 0. \overline{142857}$ , find the bar representation of $\frac{2}{7}$ .
☐	1.3 Real Numbers and their Decimal Expansions	Express $0.4 \overline{36}$ in the form $\frac{p}{q}$ .
☐	1.5 Laws of Exponents for Real Numbers	Simplify using laws of exponents: ${(\frac{81}{16})}^{\frac{- 3}{4}} \times {(\frac{25}{9})}^{\frac{- 3}{2}}$
☐	1.2 Irrational Numbers	Find two irrational numbers between √3 and √4.

CHAPTER 2: POLYNOMIALS

✓	Topic	Question / Concept
☐	2.3 Zeroes of a Polynomial	Find the remainder when $x^{3} - a x^{2} + 6 x - a$ is divided by $x - a$ .
☐	2.4 Factorisation of Polynomials	Using the Factor Theorem, show that $(x - 2)$ is a factor of $x^{3} - 2 x^{2} - 9 x + 18$ . Hence, factorise the polynomial.
☐	2.4 Factorisation of Polynomials	Factorise: $6 x^{2} + 5 \sqrt{2} x - 4$
☐	2.5 Algebraic Identities	Expand using suitable identity: $(2 x - y + 3 z)^{2}$
☐	2.5 Algebraic Identities	Factorise: $27 x^{3} + y^{3} + z^{3} - 9 x y z$
☐	2.5 Algebraic Identities	If $x + y + z = 0$ , show that $x^{3} + y^{3} + z^{3} = 3 x y z$ .
☐	2.5 Algebraic Identities	Evaluate without direct multiplication: $103 \times 107$ (using identity)
☐	2.3 Zeroes of a Polynomial	Find the value of $k$ if $x - 1$ is a factor of $p (x) = k x^{2} - 3 x + k$ .
☐	2.5 Algebraic Identities	If $a + b + c = 9$ and $a b + b c + c a = 26$ , find $a^{2} + b^{2} + c^{2}$ .

CHAPTER 3: COORDINATE GEOMETRY

✓	Topic	Question / Concept
☐	3.2 Cartesian System	In which quadrant or on which axis do the points (-2, 4), (3, -1), (-1, 0), (1, 2) lie?
☐	3.2 Cartesian System	Plot the points (4, 2), (-2, 2), (-3, -3) and (5, -3) on a graph paper. Join them in order. Name the figure obtained.
☐	3.2 Cartesian System	Write the coordinates of the vertices of a rectangle whose length and breadth are 5 and 3 units respectively, with one vertex at the origin, the longer side on the x-axis, and the rectangle lying in the third quadrant.
☐	3.2 Cartesian System	If the y-coordinate of a point is zero, where does the point lie?
☐	3.2 Cartesian System	Find the mirror image of the point (-4, 3) with respect to the x-axis.

CHAPTER 4: LINEAR EQUATIONS IN TWO VARIABLES

✓	Topic	Question / Concept
☐	4.3 Solution of a Linear Equation	Write four solutions for the equation: $2 x + y = 7$ .
☐	4.3 Solution of a Linear Equation	Draw the graph of $x + y = 7$ and $2 x - 3 y = 9$ . Find the point where they intersect.
☐	4.3 Solution of a Linear Equation	If the point (3, 4) lies on the graph of the equation $3 y = a x + 7$ , find the value of a.
☐	4.2 Linear Equations	Give the geometric representations of $y = 3$ as an equation in (a) one variable (b) two variables.
☐	4.2 Linear Equations	The cost of a notebook is twice the cost of a pen. Write a linear equation in two variables to represent this statement. (Take cost of notebook as x and cost of pen as y).
☐	4.3 Solution of a Linear Equation	Check which of the following are solutions of the equation $x - 2 y = 4$ : (i) (0, 2) (ii) (4, 0) (iii) (√2, 4√2)
☐	4.3 Solution of a Linear Equation	For what value of k, the pair of linear equations $x + 2 y = 7$ and $2 x + 4 y = k$ have infinitely many solutions?
☐	4.3 Solution of a Linear Equation	Draw the graph of the linear equation whose solutions are represented by the points having the sum of coordinates as 10 units.

CHAPTER 5: INTRODUCTION TO EUCLID'S GEOMETRY

✓	Topic	Question / Concept
☐	5.2 Euclid's Definitions, Axioms and Postulates	State Euclid's fifth postulate. How would you rewrite it in your own words?
☐	5.2 Euclid's Definitions, Axioms and Postulates	Prove that an equilateral triangle can be constructed on any given line segment.
☐	5.2 Euclid's Definitions, Axioms and Postulates	If a point C lies between two points A and B such that AC = BC, then prove that $A C = \frac{1}{2} A B$ . Explain by drawing the figure.
☐	5.2 Euclid's Definitions, Axioms and Postulates	What is a theorem? How is it different from an axiom?
☐	5.2 Euclid's Definitions, Axioms and Postulates	In the given figure, if AC = BD, then prove that AB = CD.

CHAPTER 6: LINES AND ANGLES

✓	Topic	Question / Concept
☐	6.4 Pairs of Angles	Find the measure of an angle, if six times its complement is 12° less than twice its supplement.
☐	6.5 Lines Parallel to the Same Line	In the given figure, $ℓ ∥ m$ and $t$ is a transversal. If $∠ 1 = 120^{\circ}$ , find all other angles.
☐	6.5 Lines Parallel to the Same Line	Prove that "If a transversal intersects two parallel lines, then each pair of interior angles on the same side of the transversal is supplementary."
☐	6.5 Lines Parallel to the Same Line	In the given figure, $A B ∥ C D$ . Find the value of x.
☐	6.5 Lines Parallel to the Same Line	In the given figure, $P Q ∥ R S$ , find the measure of $∠ Q R S$ .
☐	6.4 Pairs of Angles	The angles of a triangle are in the ratio 2:3:4. Find the angles.
☐	6.4 Pairs of Angles	In a $△ A B C$ , if $∠ A = 95^{\circ}$ and $∠ B = 40^{\circ}$ , find $∠ C$ .
☐	6.4 Pairs of Angles	Prove that the sum of all angles of a triangle is 180°.

CHAPTER 7: TRIANGLES

✓	Topic	Question / Concept
☐	7.3 Criteria for Congruence of Triangles	In the given figure, $A B = A C$ and $A D$ is the bisector of $∠ B A C$ . Prove that $△ A B D ≅ △ A C D$ .
☐	7.4 Some Properties of a Triangle	Prove that angles opposite to equal sides of an isosceles triangle are equal.
☐	7.4 Some Properties of a Triangle	In a $△ A B C$ , if $∠ A = ∠ C$ and $A B = 4$ cm and $A C = 6$ cm, find the length of BC.
☐	7.4 Some Properties of a Triangle	In the given figure, $P R > P Q$ and PS bisects $∠ Q P R$ . Prove that $∠ P S R > ∠ P S Q$ .
☐	7.4 Some Properties of a Triangle	Prove that the sum of any two sides of a triangle is greater than the third side.
☐	7.4 Some Properties of a Triangle	ABC is an isosceles triangle with $A B = A C$ . Draw $A P ⊥ B C$ . Show that $∠ B = ∠ C$ .
☐	7.5 Some More Criteria for Congruence of Triangles	In right triangle ABC, right-angled at C, M is the mid-point of hypotenuse AB. C is joined to M and produced to a point D such that DM = CM. Point D is joined to point B. Show that: (i) $△ A M C ≅ △ B M D$ (ii) $∠ D B C = 90^{\circ}$

CHAPTER 8: QUADRILATERALS

✓	Topic	Question / Concept
☐	8.2 Properties of a Parallelogram	Prove that the diagonals of a parallelogram bisect each other.
☐	8.2 Properties of a Parallelogram	Show that the diagonals of a square are equal and bisect each other at right angles.
☐	8.2 Properties of a Parallelogram	ABCD is a rhombus. Show that diagonal AC bisects $∠ A$ and $∠ C$ .
☐	8.3 The Mid-point Theorem	Prove the Midpoint Theorem: The line segment joining the mid-points of two sides of a triangle is parallel to the third side and equal to half of it.
☐	8.3 The Mid-point Theorem	ABC is a triangle right-angled at C. A line through the mid-point M of hypotenuse AB and parallel to BC intersects AC at D. Show that D is the mid-point of AC.
☐	8.2 Properties of a Parallelogram	ABCD is a trapezium with $A B ∥ D C$ . E and F are points on non-parallel sides AD and BC respectively such that EF is parallel to AB. Show that $\frac{A E}{E D} = \frac{B F}{F C}$ .
☐	8.2 Properties of a Parallelogram	In a parallelogram, show that the angle bisectors of two adjacent angles intersect at a right angle.

CHAPTER 9: CIRCLES

✓	Topic	Question / Concept
☐	9.2 Angle Subtended by a Chord at a Point	Prove that equal chords of a circle subtend equal angles at the centre.
☐	9.3 Perpendicular from the Centre to a Chord	Prove that the perpendicular from the centre of a circle to a chord bisects the chord.
☐	9.4 Equal Chords and their Distances from the Centre	Prove that equal chords of a circle are equidistant from the centre.
☐	9.5 Angle Subtended by an Arc of a Circle	Prove that the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.
☐	9.5 Angle Subtended by an Arc of a Circle	Prove that angles in the same segment of a circle are equal.
☐	9.6 Cyclic Quadrilaterals	Prove that the sum of either pair of opposite angles of a cyclic quadrilateral is 180°.
☐	9.6 Cyclic Quadrilaterals	Prove that the quadrilateral formed by angle bisectors of a cyclic quadrilateral is also cyclic.
☐	9.5 Angle Subtended by an Arc of a Circle	In the given figure, A, B, C, D are four points on a circle. AC and BD intersect at E such that $∠ B E C = 130^{\circ}$ and $∠ E C D = 20^{\circ}$ . Find $∠ B A C$ .
☐	9.4 Equal Chords and their Distances from the Centre	If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to corresponding segments of the other chord.
☐	9.5 Angle Subtended by an Arc of a Circle	A circular park of radius 20 m is situated in a colony. Three boys Ankur, Syed and David are sitting at equal distance on its boundary each having a toy telephone in his hands to talk each other. Find the length of the string of each phone.

CHAPTER 10: HERON'S FORMULA

✓	Topic	Question / Concept
☐	10.1 Area of a Triangle – by Heron's Formula	Find the area of a triangle whose sides are 8 cm, 11 cm, and 13 cm.
☐	10.1 Area of a Triangle – by Heron's Formula	An isosceles triangle has perimeter 30 cm and each of the equal sides is 12 cm. Find the area of the triangle.
☐	10.1 Area of a Triangle – by Heron's Formula	The sides of a triangular plot are in the ratio 3:5:7 and its perimeter is 300 m. Find its area.
☐	10.1 Area of a Triangle – by Heron's Formula	Using Heron's formula, find the area of an equilateral triangle whose side is 'a' cm.
☐	10.1 Area of a Triangle – by Heron's Formula	A traffic signal board, indicating 'SCHOOL AHEAD', is an equilateral triangle with side 'a'. Find the area of the signal board, using Heron's formula. If its perimeter is 180 cm, what will be the area of the signal board?
☐	10.1 Area of a Triangle – by Heron's Formula	The edges of a triangular board are 6 cm, 8 cm and 10 cm. Calculate the cost of painting it at the rate of 9 paise per cm².
☐	10.1 Area of a Triangle – by Heron's Formula	A field is in the shape of a trapezium whose parallel sides are 25 m and 10 m. The non-parallel sides are 14 m and 13 m. Find the area of the field.

CHAPTER 11: SURFACE AREAS AND VOLUMES

✓	Topic	Question / Concept
☐	11.1 Surface Area of a Right Circular Cone	The curved surface area of a right circular cylinder of height 14 cm is 88 cm². Find the diameter of the base of the cylinder.
☐	11.1 Surface Area of a Right Circular Cone	Find the total surface area of a cone, if its slant height is 21 m and diameter of its base is 24 m.
☐	11.2 Surface Area of a Sphere	Find the volume of a sphere whose surface area is 154 cm².
☐	11.2 Surface Area of a Sphere	A hemispherical bowl has a radius of 3.5 cm. What would be the volume of water it would contain?
☐	11.1 Surface Area of a Right Circular Cone	A wall of length 10 m was to be built across an open ground. The wall has 10 m length, 40 cm width and 5 m height. If the wall is built using bricks of dimensions 20 cm × 10 cm × 8 cm, how many bricks would be required?
☐	11.2 Surface Area of a Sphere	The radius of a spherical balloon increases from 7 cm to 14 cm as air is being pumped into it. Find the ratio of the surface areas of the balloon in the two cases.
☐	11.1 Surface Area of a Right Circular Cone	A joker's cap is in the form of a right circular cone of base radius 7 cm and height 24 cm. Find the area of the sheet required to make 10 such caps.
☐	11.2 Surface Area of a Sphere	A dome of a building is in the form of a hemisphere. From inside, it was white-washed at the cost of ₹4989.60. If the cost of white-washing is ₹20 per square metre, find the (i) inside surface area of the dome, (ii) volume of the air inside the dome.
☐	11.3 Volume of a Right Circular Cone	A right triangle with sides 6 cm, 8 cm and 10 cm is revolved about the side 8 cm. Find the volume and curved surface area of the solid so formed.

CHAPTER 12: STATISTICS

✓	Topic	Question / Concept
☐	12.1 Graphical Representation of Data	The marks obtained by 30 students of Class IX in a test are given. Construct a frequency distribution table with class intervals of equal width, say 0-10, 10-20, etc.
☐	12.1 Graphical Representation of Data	Draw a histogram for the given data.
☐	12.1 Graphical Representation of Data	Draw a frequency polygon for the given data, without using a histogram.
☐	12.1 Graphical Representation of Data	Find the mean of the first five natural numbers.
☐	12.1 Graphical Representation of Data	The mean of 10 observations was found to be 40. Later on, it was detected that an observation 21 was misread as 12. Find the correct mean.
☐	12.1 Graphical Representation of Data	The following observations are arranged in ascending order. If the median of the data is 63, find the value of x: 29, 32, 48, 50, x, x+2, 72, 78, 84, 95
☐	12.1 Graphical Representation of Data	Find the mode of the following data: 15, 18, 16, 15, 14, 18, 19, 15, 17, 16.
☐	12.1 Graphical Representation of Data	Making Data Continuous: Convert the following discontinuous classes into continuous classes: 10-19, 20-29, 30-39
☐	12.1 Graphical Representation of Data	Adjusted Frequency: Calculate the adjusted frequencies for the following data to draw a histogram (minimum class width = 10): Class: 0-10 (f=5), 10-20 (f=8), 20-40 (f=18), 40-60 (f=12)

How to Use This Checklist:

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Tick the box (✓) when you have completed a question.
Revise the ticked questions before exams.
Practice unticked questions until you master them.

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CBSE Class 9 Maths Checklist

Complete chapter-wise topics with checkboxes • PDF • 12 chapters

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Sunday, February 15, 2026