Grade 10
  1. Number Systems  1.1. Real Numbers  1.1.1. Properties of Real Numbers  1.1.2. Rational and Irrational Numbers  1.1.3. Decimal Representation of Real Numbers  1.1.4. Operations on Real Numbers  1.1.5. Real Numbers on the Number Line  1.2. Euclid's Division Lemma  1.3. Prime Factorization  1.4. LCM and HCF  1.5. Exponents and Radicals  1.5.1. Laws of Exponents  1.5.2. Simplifying Radical Expressions  1.5.3. Scientific Notation
  2. Algebra  2.1. Polynomials  2.1.1. Definition and Types of Polynomials  2.1.2. Degree of a Polynomial  2.1.3. Zeros of a Polynomial  2.1.4. Factorization of Polynomials  2.1.5. Algebraic Identities  2.2. Linear Equations in Two Variables  2.2.1. Graph of Linear Equations  2.2.2. Solutions of Linear Equations  2.2.3. Application of Linear Equations in Word Problems  2.3. Quadratic Equations  2.3.1. Standard Form of a Quadratic Equation  2.3.2. Methods of Solving Quadratic Equations  2.3.2.1. Factorization Method  2.3.2.2. Completing the Square Method  2.3.2.3. Quadratic Formula  2.3.3. Nature of Roots  2.4. Arithmetic Progressions  2.4.1. Definition of Arithmetic Progression  2.4.2. General Term of an AP  2.4.3. Sum of First n Terms of an AP  2.5. Introduction to Functions  2.5.1. Definition and Types of Functions  2.5.2. Domain and Range  2.5.3. Composition of Functions  2.5.4. Inverse of a Function
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Distance Formula  3.3. Section Formula  3.4. Area of a Triangle  3.5. Slope of a Line  3.6. Equation of a Line  3.6.1. Slope-Intercept Form  3.6.2. Point-Slope Form  3.6.3. Two-Point Form  3.6.4. Intercept Form  3.6.5. General Form of a Line
  4. Trigonometry  4.1. Trigonometric Ratios  4.1.1. Sine Cosine Tangent and their Reciprocals  4.1.2. Values of Trigonometric Ratios at Specific Angles  4.2. Trigonometric Identities  4.3. Trigonometric Ratios of Complementary Angles  4.4. Heights and Distances  4.4.1. Angle of Elevation and Depression  4.4.2. Applications in Real-Life Problems
  5. Geometry  5.1. Triangles  5.1.1. Congruence of Triangles  5.1.2. Criteria for Congruence  5.1.3. Properties of Triangles  5.2. Similarity  5.2.1. Similar Figures  5.2.2. Criteria for Similarity  5.2.3. Similarity of Triangles  5.2.4. Applications of Similarity in Real-Life  5.3. Circles  5.3.1. Properties of a Circle  5.3.2. Tangent to a Circle  5.3.3. Number of Tangents from a Point  5.3.4. Angle Subtended by an Arc  5.4. Constructions  5.4.1. Construction of Similar Triangles  5.4.2. Tangent to a Circle  5.4.3. Construction of Angle Bisectors
  6. Mensuration  6.1. Areas of Plane Figures  6.1.1. Area of a Triangle  6.1.2. Area of a Parallelogram  6.1.3. Area of a Trapezium  6.1.4. Area of a Rhombus  6.2. Surface Areas and Volumes  6.2.1. Surface Area of a Cube Cuboid and Cylinder  6.2.2. Surface Area of a Cone and Sphere  6.2.3. Volume of a Cube Cuboid and Cylinder  6.2.4. Volume of a Cone and Sphere  6.3. Conversion of Units  6.4. Frustum of a Cone
  7. Statistics  7.1. Introduction to Statistics  7.2. Collection of Data  7.3. Presentation of Data  7.3.1. Tabular Form  7.3.2. Graphical Form  7.3.2.1. Bar Graph  7.3.2.2. Histogram  7.3.2.3. Frequency Polygon  7.3.2.4. Ogive  7.4. Measures of Central Tendency  7.4.1. Mean Median and Mode  7.4.2. Quartiles and Percentiles  7.5. Measures of Dispersion  7.5.1. Range and Interquartile Range  7.5.2. Standard Deviation and Variance
  8. Probability  8.1. Introduction to Probability  8.2. Experimental Probability  8.3. Theoretical Probability  8.4. Probability of Simple Events  8.5. Complementary Events  8.6. Probability of Compound Events

Grade 10 ↓


Geometry


Geometry is a branch of mathematics that deals with the properties of shapes, sizes, and space. It is a subject that is both practical and aesthetic, helping us understand the world around us. In grade 10 maths, geometry introduces a number of topics that involve the study of various geometric shapes, theorems, and concepts. An understanding of these topics is useful in various fields such as engineering, architecture, art, and design.

Basic shapes and definitions

Geometry begins with understanding some basic shapes and their definitions. These form the basis for more complex studies.

  • Point: A point is a location in space. It has no size, width, length, and depth. It is represented by a dot and is usually designated by a letter, for example, A
  • Line: A line is a collection of points that extends infinitely in two directions. It has length but no thickness and is usually represented by two points along the line, such as Line AB.
  • Line segment: A portion of a line bounded by two distinct endpoints, and containing every point on the line between its endpoints. Example: Segment CD.
  • Ray: A ray is a part of a line that starts from a point and goes to infinity in a particular direction. It is represented as Ray EF.
  • Angle: An angle is formed by two rays (sides of the angle) that have a common endpoint (vertex). It is measured in degrees. Example: <GHI = 45°.

Triangle

A triangle is a polygon with three sides. It is one of the simplest shapes in geometry. The study of triangles involves understanding the types of triangles and their properties.

Types of triangles according to sides

  • Equilateral triangle: All three sides are of equal length. Each interior angle is 60 degrees.
  • Isosceles triangle: It has two sides of equal length. The angles opposite to these sides are also equal.
  • Scalene triangle: All sides are different lengths, and all angles are different.

Types of triangles based on angles

  • Acute triangle: All angles are less than 90 degrees.
  • Right triangle: It has one angle of 90 degrees.
  • Obtuse triangle: One of its angles is more than 90 degrees.

Pythagorean theorem

The Pythagorean theorem is a fundamental principle in geometry, particularly useful in understanding right triangles. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be expressed by the formula:

a 2 + b 2 = c 2

where c is the hypotenuse, and a and b are the other two sides.

Quadrilateral

Quadrilaterals are polygons with four sides and four vertices. They come in different forms, each of which has different properties.

Common types of quadrilaterals

  • Square: A quadrilateral with all sides of equal length and all angles equal to 90 degrees.
  • Rectangle: Opposite sides are equal in length, and all angles are 90 degrees. This is essentially an "elongated square."
  • Rhombus: All sides are the same length, like a square, but the angles are not necessarily 90 degrees.
  • Parallelogram: Opposite sides are parallel and of equal length, and opposite angles are also equal.
  • Trapezoid (or trapezium in British English): has only one pair of parallel sides.

Circles

A circle is a circular figure that has no corners or edges. It is defined as the set of all points in a plane that are at a fixed distance from the center point.

Parts of a circle

  • Radius: The distance from the center of a circle to any point on the circle.
  • Diameter: The line that passes through the center of a circle and has its endpoints on the circle. It is twice the length of the radius.
  • Circumference: The distance around a circle. It is calculated using the formula C = 2 π r, where r is the radius.
  • Arc: A portion of the circumference of a circle.
  • Chord: A line segment whose endpoints lie on the circle.
  • Tangent: A line that touches a circle at exactly one point.

A circle can also be represented by this equation: (x - h) 2 + (y - k) 2 = r 2, where (h, k) is the center and r is the radius.

Perimeter and area

Perimeter refers to the total distance around a shape, while area is a measure of the space occupied by a shape. Here are basic formulas to calculate the perimeter and area of some common shapes.

Perimeter formula

  • Rectangle: 2(length + width)
  • Square: 4 × side
  • Triangle: a + b + c, where a, b, c are the lengths of the sides
  • Circle (circumference): 2πr

Area formula

  • Rectangle: length × width
  • Category: side 2
  • Triangle: 0.5 × base × height
  • Circle: πr 2

Congruence and similarity

In geometry, the concepts of congruence and similarity are important for understanding how shapes relate to one another.

Similar shapes

Similar figures have the same shape, but not necessarily the same size. Their sides are proportional and the angles are equal. If triangle ABC is similar to triangle DEF, then AB/DE = BC/EF = CA/FD, and the corresponding angles are equal.

Transformations

Transformations in geometry involve moving or changing a shape in some way, while maintaining its integrity. There are four main types of transformations.

Types of changes

  • Translation: This involves moving a shape from one place to another without rotating or flipping it.
  • Rotation: In this, a figure is rotated about a fixed point called the centre of rotation.
  • Reflection: This involves flipping a figure on a line to form a mirror image.
  • Scaling: This involves changing the size of a shape by making it bigger or smaller, while keeping its proportions intact.

Coordinate geometry

Coordinate geometry, also called analytical geometry, is the study of geometry using coordinate systems. It involves using algebra to understand and solve geometric problems.

Cartesian plane

The Cartesian plane is a two-dimensional coordinate system defined by a horizontal line known as the x-axis and a vertical line called the y-axis. The point where the two axes intersect is the origin, which has coordinates (0, 0).

Slope of the line

The slope of a line in the Cartesian plane is a measure of its steepness and direction. It is calculated as follows:

m = (y 2 - y 1) / (x 2 - x 1)

where (x 1, y 1) and (x 2, y 2) are any two points on the line.

Equation of line

The equation of a line can be written in different forms, such as:

Slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept.

Standard form: Ax + By = C, where A, B, and C are constants.

Point-slope form: y - y 1 = m(x - x 1), where (x 1, y 1) is a point on the line and m is the slope.

Conclusion

Geometry is a vast and fascinating field that provides us with the tools to view and understand the world in a mathematical way. From understanding basic shapes to solving complex problems, the principles of geometry are extremely useful and important. By mastering geometry, you gain a foundation that supports further study in mathematics and various real-world applications.

A good understanding of geometry is very important in Class 10 Maths as it sets the stage for more advanced studies and applications in higher education and various career paths. The concepts covered here, though elementary, are part of a structured learning approach to understanding more complex geometric problems and theorems that are encountered in later studies.


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