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 ↓


Triangles


Triangles are one of the fundamental shapes in geometry. A triangle is a three-sided polygon, meaning it is a closed shape with three edges and three vertices. The study of triangles is important because they are the simplest type of polygon and are used in various aspects such as architecture, engineering, astronomy, and art.

Types of triangles

Triangles can be classified into different types based on their angles and the lengths of their sides. Understanding these types is important for identifying and solving problems involving triangles.

Based on the length of the sides

There are three main types of triangles depending on the length of the sides:

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

Based on angles

The types of triangles based on the angles are as follows:

  • Acute triangle: All three interior angles are less than 90 degrees.
  • Right triangle: One of these angles is 90 degrees. The side opposite this angle is the longest and is called the hypotenuse.
  • Obtuse triangle: One of its angles is more than 90 degrees.

Properties of triangles

Triangles have many interesting properties, such as:

Sum of interior angles

The sum of the interior angles of any triangle is always 180 degrees. If the angles in a triangle are A, B and C, then this property can be expressed as:

A + B + C = 180°

Visual example

Consider a simple acute triangle represented visually:

A B C

In this triangle, the sum of angles A, B and C is 180 degrees.

Pythagorean theorem

In right triangles, the Pythagorean theorem is a fundamental relation expressed as:

a² + b² = c²

where a and b are the lengths of the two shorter sides, and c is the length of the hypotenuse.

Visual example

A B C

Congruence and similarity

Congruent triangles

Two triangles are congruent if all corresponding sides and angles are equal. The most commonly used congruence criteria are:

  • SSS (Side-Side-Side): If three sides of a triangle are equal to three sides of another triangle.
  • SAS (Side-Angle-Side): If two sides and their included angle of a triangle are equal to two sides and their included angle of another triangle.
  • ASA (Angle-Side-Angle): If two angles and the included side of a triangle are equal to two angles and the included side of another triangle.
  • AAS (Angle-Angle-Side): If two angles and a side of a triangle are equal to the corresponding two angles and a side of another triangle.
  • RHS (Right Angle-Hypotenuse-Side): In right-angled triangles, if the hypotenuse and a side of one triangle are equal to the hypotenuse and a side of the other triangle.

Similar triangles

Triangles are similar if their corresponding angles are equal, and their corresponding sides are proportional. The common similarity criteria are:

  • AA (Angle-Angle): If two angles of a triangle are equal to two angles of another triangle.
  • SAS (Side-Angle-Side): If the angle between two sides of a triangle is equal to the angle between two sides of another triangle, and the sides are in the same proportion.
  • SSS (Side-Side-Side): If the corresponding sides of two triangles are in proportion.

Critical centers in a triangle

There are several important points associated with triangles, known as centers. These include the centroid, incenter, circumcenter, and orthocenter.

Centroid

The centroid is the point where the three medians of a triangle intersect. The median is a line drawn from a vertex to the midpoint of the opposite side. The centroid divides each median into two segments, one of which is twice as long as the other.

Visual example

The red dot is the centroid.

Incenter

The incenter is the intersection point of the triangle's angle bisectors. It is the center of the incircle, which is the largest circle that fits inside the triangle.

Circumcenter

The circumcenter is the point where the perpendicular bisectors of the sides intersect. It is the center of the circumcircle, which passes through all the vertices of the triangle.

Orthocenter

The orthocenter is the point where the altitudes (or extensions of altitudes) of a triangle intersect each other. An altitude is a perpendicular line from a vertex to the line on the opposite side.

Area and perimeter of a triangle

Calculating the area and perimeter of a triangle is a fundamental skill in geometry.

Perimeter

The perimeter of a triangle is simply the sum of the lengths of its sides. If the lengths of the sides of a triangle are a, b and c, then the perimeter P is given by:

P = a + b + c

Area

The area of a triangle can be calculated using different formulas depending on the known information:

  • Base-height formula: If the base b and the height h are known, then the area A is: 
    A = (1/2) × b × h
  • Heron's formula: If all three sides are known, use Heron's formula for the area: 
    s = (a + b + c) / 2
A = √(s × (s - a) × (s - b) × (s - c))
    where s is the semiperimeter.

Special right triangles

Some right triangles have special properties that make them particularly useful.

45-45-90 triangle

The angles in this triangle are 45 degrees, 45 degrees and 90 degrees. The sides are in the ratio 1:1:√2. Therefore, if the legs are x, then the hypotenuse will be x√2.

Visual example

X X x√2

30-60-90 triangle

In a 30-60-90 triangle, the sides are in the ratio 1:√3:2. If the shortest side (opposite the 30 degree angle) is x, then the side opposite the 60 degree angle will be x√3, and the hypotenuse will be 2x.

Visual example

X x√3 2x

Applications of triangles

Triangles are used in various applications such as trigonometry, navigation, computer graphics and structural engineering. Understanding the properties and types of triangles can help in solving complex problems related to them.

Example problems

Let us look at some example problems to understand the concepts better:

Example 1

Given a right triangle with sides 3, 4, and 5, verify whether it is a right triangle.

Calculate using Pythagorean theorem:
3² + 4² = 9 + 16 = 25
5² = 25
Since both sides of the equation are equal, it confirms that the triangle is a right triangle.

Example 2

Find the area of an equilateral triangle whose side length is 6 units.

The formula for the area of an equilateral triangle is:
A = (√3 / 4) × a²
A = (√3 / 4) × 6² = 9√3 square units.

Grade 10 → 5.1


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