Class 10 Notes - Triangles

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Triangles

A triangle is a closed figure formed by three line segments. It is one of the most fundamental shapes in geometry and has unique properties and classifications based on its sides and angles.

Types of Triangles

  • By Sides:
    • Equilateral Triangle: All three sides are equal, and all angles are 60°.
    • Isosceles Triangle: Two sides are equal, and the angles opposite these sides are equal.
    • Scalene Triangle: All sides and all angles are different.
  • By Angles:
    • Acute-angled Triangle: All angles are less than 90°.
    • Right-angled Triangle: One angle is exactly 90°.
    • Obtuse-angled Triangle: One angle is greater than 90°.

Properties of Triangles

  • The sum of the interior angles of a triangle is always 180°.
  • The sum of the lengths of any two sides is greater than the length of the third side (Triangle Inequality Theorem).
  • The exterior angle of a triangle is equal to the sum of the two opposite interior angles.

Area of a Triangle

  • Standard Formula:
    Area = (1/2) × base × height
  • Heron's Formula:
    If the sides are a, b, c:
    s = (a + b + c) / 2
    Area = √[s(s - a)(s - b)(s - c)]

Pythagoras Theorem (for Right-angled Triangles)

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
c2 = a2 + b2

Congruence of Triangles

  • SSS (Side-Side-Side)
  • SAS (Side-Angle-Side)
  • ASA (Angle-Side-Angle)
  • AAS (Angle-Angle-Side)
  • RHS (Right angle-Hypotenuse-Side)

Similarity of Triangles

  • AA (Angle-Angle)
  • SSS (Side-Side-Side)
  • SAS (Side-Angle-Side)

Important Points

  • Centroid: Intersection of medians.
  • Incentre: Intersection of angle bisectors.
  • Orthocentre: Intersection of altitudes.
  • Circumcentre: Intersection of perpendicular bisectors.

Word Problems

  1. Find the area of a triangle with base 8 cm and height 5 cm.
  2. If the sides of a triangle are 7 cm, 8 cm, and 9 cm, find its area using Heron's formula.
  3. In a right-angled triangle, if one side is 6 cm and the other is 8 cm, find the hypotenuse.
  4. Prove that two triangles are congruent if their three sides are equal.

Summary

  • Triangles are three-sided polygons with important properties and formulas.
  • They are classified by sides and angles.
  • Area can be found using different formulas.
  • Congruence and similarity help in comparing triangles.