Class 9 Mathematic Triangles All Formula & Concept Notes | Formula Sheet

Class 9 Mathematics: Introduction to Triangles - Concepts and Formulas

Triangles are a fundamental topic in geometry and understanding them is crucial as they form the basis of many other concepts in mathematics. Here's a detailed breakdown of the key concepts, properties, and formulas related to triangles as introduced in Class 9.

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1. Basic Definitions and Terminology

Triangle:

A triangle is a three-sided polygon with three edges and three vertices.

Vertices, Edges, and Angles:

• Vertices are the points where two sides of a triangle meet. A triangle has three vertices.
• Edges (or sides) are the straight lines that connect the vertices.
• Angles are the spaces between two sides and are measured in degrees.

Types of Triangles:

Based on Sides:

• Scalene Triangle: All three sides are of different lengths.
• Isosceles Triangle: Two sides are of equal length.
• Equilateral Triangle: All three sides are of equal length.

Based on Angles:

• Acute Triangle: All three angles are less than 90°.
• Right Triangle: One angle is exactly 90°.
• Obtuse Triangle: One angle is more than 90°.

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2. Angle Sum Property of a Triangle

Theorem:

The sum of the interior angles of a triangle is always 180°.

$$\text{∠A + ∠B + ∠C = 180°}$$

This is known as the Angle Sum Property.

Exterior Angle Property:

The exterior angle of a triangle is equal to the sum of the two opposite interior angles.

$$\text{∠Exterior = ∠Interior1 + ∠Interior2}$$

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3. Types of Triangles Based on Angle Measures

Right-Angled Triangle:

A triangle with one angle equal to 90°.

Isosceles Right Triangle:

A right-angled triangle where the two legs (sides forming the right angle) are equal in length.

Properties of Right-Angled Triangle:

• Pythagoras Theorem: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

$$\text{Hypotenuse}^2 = \text{Base}^2 + \text{Perpendicular}^2$$

Acute Triangle:

A triangle where all angles are less than 90°.

Obtuse Triangle:

A triangle where one angle is greater than 90°.

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4. Congruence of Triangles

Congruent Triangles:

Two triangles are said to be congruent if all their corresponding sides and angles are equal.

Criteria for Congruence:

1. SSS (Side-Side-Side): If three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent.

2. SAS (Side-Angle-Side): If two sides and the angle between them in one triangle are equal to the corresponding sides and angle in another triangle, then the triangles are congruent.

3. ASA (Angle-Side-Angle): If two angles and the side between them in one triangle are equal to the corresponding angles and side in another triangle, then the triangles are congruent.

4. AAS (Angle-Angle-Side): If two angles and the side opposite one of these angles in one triangle are equal to the corresponding angles and side in another triangle, then the triangles are congruent.

5. RHS (Right Angle-Hypotenuse-Side): In right-angled triangles, if the hypotenuse and one side of a triangle are equal to the hypotenuse and one side of another triangle, then the triangles are congruent.

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5. Inequality in Triangles

Triangle Inequality Theorem:

In any triangle, the sum of the lengths of any two sides is greater than the length of the third side.

$$\text{AB + BC > AC}$$

Angle-Side Relationships:

• The side opposite the larger angle is longer.
• Conversely, the angle opposite the longer side is larger.

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6. Medians, Altitudes, and Perpendicular Bisectors

Median:

A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side.

Altitude:

An altitude of a triangle is a perpendicular drawn from a vertex to the opposite side (or its extension).

Perpendicular Bisector:

A perpendicular bisector of a side of a triangle is a line perpendicular to the side and passing through its midpoint.

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7. Mid-Point Theorem

Theorem:

The line segment joining the midpoints of two sides of a triangle is parallel to the third side and half of it.

$$\text{If D and E are midpoints of AB and AC respectively, then DE || BC and DE = ½ BC.}$$

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8. Heron’s Formula (Area of a Triangle)

Formula:

The area of a triangle can be calculated using Heron’s formula if the lengths of all three sides are known.

$$\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$$

Where:

• $s = \frac{a+b+c}{2}$ is the semi-perimeter.
• $a, b, c$ are the sides of the triangle.

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These are the core concepts and formulas related to triangles as introduced in Class 9 Mathematics. Understanding these basics is essential for progressing to more advanced geometric topics in later grades.