Study Tips: Properties of Triangles That You Need To Know

This Coordinate Geometry tip was contributed by Kshitij of BITS Pilani, Goa Campus.

What you will learn from this tip:
-- Formulae to calculate trigonometric ratios of half-angles
-- The properties of every triangle

Trigonometric ratios of half-angles

Here are some important formulae that you should note down to calculate the trigonometric ratios of half-angles.

  1. sin A/2 = √(s-b)(s-c) / bc
  2. sin B/2 = √(s-b)(s-c) / bc
  3. sin C/2 = √(s-a)(s-b) / ab
  4. cos A/2 = √s(s-a) / bc
  5. cos B/2 = √s(s-b) / ca
  6. cos C/2 = √s(s-c) / ab
  7. tan A/2 = √(s-b)(s-c) / s(s-a)
  8. tan B/2 = √(s-c)(s-a) / s(s-b)
  9. tan C/2 = √(s-a)(s-b) / s(s-c)

where s is semi-perimeter and is calculated as s= (a+b+c)/2 and, a, b and c are sides of a triangle.

The 5 Properties Of A Triangle

  1. For all polygons including the triangle, the sum of exterior angles is 360 degrees.

  2. For a triangle ABC, the following 2 conditions are sufficient to prove that it is a triangle: (1) Angles (CAB + ABC + BCA) = 180 degrees (2) The sum of the length of any two sides is greater than other side.

  3. To know the complete properties of Triangleut of 3 Angles and 3 Sides, one must know any 3 measurements.

  4. Angle-Side-Angle (ASA), Side-Angle-Side (SAS) and Side-Side-Side (SSS) tests are sufficient to prove two triangles are equal. AAS is not a test as such. For right angled triangle, hypotenuse side test is sufficient.

  5. The sum of two interior angles is equal to the exterior angle of the rest of the angle.

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