Quick Revision Notes

Class: 7 Mathematics

Chapter: The Triangle and its Properties

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1. Introduction to a Triangle

A triangle is a simple closed curve made of three line segments. It has:

• 3 Sides: AB, BC, CA
• 3 Angles: ∠A, ∠B, ∠C
• 3 Vertices: A, B, C

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2. Classification of Triangles

Triangles are classified into two main categories:

A. Based on Sides:

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

B. Based on Angles:

1. Acute-angled Triangle: All three angles are acute (less than 90°).
2. Right-angled Triangle: One angle is a right angle (exactly 90°).
3. Obtuse-angled Triangle: One angle is an obtuse angle (greater than 90°).

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3. Medians and Altitudes of a Triangle

• Median: A line segment joining a vertex to the mid-point of the opposite side. A triangle has 3 medians that meet at a single point called the centroid.
• Altitude (Height): A perpendicular line segment from a vertex to the opposite side. A triangle has 3 altitudes that meet at a single point called the orthocentre.

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4. Exterior Angle of a Triangle and Its Property

• Exterior Angle Property: The measure of any exterior angle of a triangle is equal to the sum of the measures of its two interior opposite angles.
  • Formula: Exterior Angle = Sum of Interior Opposite Angles

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5. Angle Sum Property of a Triangle

• Theorem: The total sum of the three interior angles of a triangle is always 180°.
  • Formula: ∠A + ∠B + ∠C = 180°

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6. Two Special Triangles: Equilateral and Isosceles

• Equilateral Triangle:
  • All sides are equal.
  • All interior angles are equal, and each measures 60°.
• Isosceles Triangle:
  • Two sides are equal.
  • The angles opposite to the equal sides are also equal.

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7. Sum of the Lengths of Two Sides of a Triangle (Triangle Inequality Property)

• Property: The sum of the lengths of any two sides of a triangle is always greater than the length of the third side.
  • Formulas:
    • Side 1 + Side 2 > Side 3
    • Side 2 + Side 3 > Side 1
    • Side 3 + Side 1 > Side 2
• Note: The difference between the lengths of any two sides is always less than the third side.

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8. Right-Angled Triangle and Pythagoras Property

In a right-angled triangle, the side opposite to the right angle is called the hypotenuse, and the other two sides are called the legs.

• 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.
  • Formula: (Hypotenuse)^2 = (Base)^2 + (Perpendicular)^2
  • Or: AC^2 = AB^2 + BC^2 (where AC is the hypotenuse).