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.

**sin A/2**= √(s-b)(s-c) / bc**sin B/2**= √(s-b)(s-c) / bc**sin C/2**= √(s-a)(s-b) / ab**cos A/2**= √s(s-a) / bc**cos B/2**= √s(s-b) / ca**cos C/2**= √s(s-c) / ab**tan A/2**= √(s-b)(s-c) / s(s-a)**tan B/2**= √(s-c)(s-a) / s(s-b)**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

- For all polygons including the triangle, the sum of exterior angles is 360 degrees.
- 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.
- To know the complete properties of Triangleut of 3 Angles and 3 Sides, one must know any 3 measurements.
- 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.
- The sum of two interior angles is equal to the exterior angle of the rest of the angle.

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