Triangle Fundamentals | In-Depth Guide to Types & Properties

Triangle Fundamentals | In-Depth Guide to Types & Properties

Triangle Fundamentals | In-Depth Guide to Types & Properties

A triangle, one of the most fundamental geometric shapes, is a polygon with three sides. It is formed by connecting three non-collinear points with straight lines. These lines—AB, BC, and CA—are the sides of the triangle, while the angles formed at these points (A, B, and C) are its angles.

In this article, we delve into the fundamentals of triangles, their properties, classifications, and applications through solved examples.


Basic Properties of Triangles

  1. Sum of Angles:
    The sum of the angles in a triangle is always 180°.
    • Example: ∠A + ∠B + ∠C = 180°.
  2. Triangle Inequality Theorem:
    The sum of any two sides of a triangle is greater than the third side.
    • Example: AB + BC > CA, BC + CA > AB, CA + AB > BC.
  3. Exterior Angle Property:
    The exterior angle of a triangle is equal to the sum of the two opposite interior angles.

Classifying Triangles

1. Based on Sides:

  • Scalene Triangle:
    All three sides are unequal.
  • Isosceles Triangle:
    Two sides are equal, and the angles opposite these sides are also equal.
  • Equilateral Triangle:
    All three sides and angles are equal, with each angle measuring 60°.
    • Length of altitude:
    • Area:

2. Based on Angles:

  • Acute-angled Triangle:
    All three angles are less than 90°.
  • Obtuse-angled Triangle:
    One angle is greater than 90°.
  • Right-angled Triangle:
    One angle is 90°. The side opposite the right angle is the hypotenuse, which is the longest side.
    • Pythagoras theorem: .

Solved Examples

Example 1:

Find the length of the base of a triangle whose altitude is 4 cm longer than the base. The area is 96 cm².

Solution:

  • Let the base = cm and altitude = cm.
    Area = .
    Solving, .
    cm (base), and altitude = 16 cm.

Example 2:

An isosceles triangle has a perimeter of 100 cm. The base is 49 cm. Find the equal sides.

Solution:

  • Let each equal side = .
    .
    cm.

Example 3:

A triangle has an area of 615 cm². One side is 123 cm. Find the perpendicular dropped on it.

Solution:

  • Area = .
    cm.

Example 4:

Find the area of a right triangle with base = 7 cm and height = 6 cm.

Solution:

  • Area = cm².

Advanced Properties and Applications

Altitude and Orthocenter:

  • Altitudes of a triangle intersect at the orthocenter.
  • For a right triangle, the orthocenter lies at the vertex of the right angle.

Inradius and Circumradius:

  • Inradius (): Radius of the incircle, tangent to all sides of the triangle.
  • Circumradius (): For a right triangle, .

Euler Line:

  • In right triangles, the Euler line passes through the centroid, orthocenter, and circumcenter.

Conclusion

Triangles form the foundation of geometry, with their principles extending into trigonometry, engineering, and beyond. Understanding their properties is crucial for solving practical problems in various fields. For more in-depth learning and hands-on practice, explore FACE Prep CRT program, which offers comprehensive training programs tailored for students aiming to excel in academics and placements.

Triangle Fundamentals

 

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