A closed figure enclosed by 3 sides is called a Triangle. ABC is a triangle. The sides AB, BC, AC are respectively denoted by c, a, b. Please carefully note the capital and small letters.

In any triangle ABC

(1) \(A = \frac{1}{2} a \times h = \frac{1}{2} base\) \(\times\) perpendicular to base from opposite vertex
(2) \(A = \sqrt{s(s –a)(s – b)(s – c)}\) , \(s = \frac{(a + b + c)}{2} =\) semi–perimeter
(3) \(P = (a + b + c) = 2 s\)


PROPERTIES :
1. Sum of the three interior angles is 180°
2. When one side is extended in any direction, an angle is formed with another side. This is called the exterior angle.
3. There are six exterior angles of a triangle.
4. Interior angle + corresponding exterior angle = 180°.
5. An exterior angle = Sum of the other two interior angles not adjacent to it
6. Sum of any two sides is greater than the third side.
7. Difference of any two sides is less than the third side.
8. Side opposite to the greatest angle will be the greatest and vice versa.
9. A triangle must have at least two acute angles.
10. Triangles on equal bases and between the same parallels have equal areas.
11. If a, b, c denote the sides of a triangle then

(i) if \(c^2 < a^2 + b^2\), Triangle is acute angled
(ii) if \(c^2 = a^2 + b^2\), Triangle is right angled
(iii) if \(c^2 > a^2 + b^2\), Triangle is obtuse angled


Right Angled Triangle



A triangle whose one angle is 90° is called a right (angled) Triangle. In the figure, b is the hypotenuse, and a & c the legs, called base and height resp.

(1) \(h^2 = m n\)
(2) \(AC^2 = AB^2 + BC^2\) (Pythagoras theorem)
(3) \(h = \frac{ac}{b}\)
(4) \(Area = \frac{ac }{ 2}\)

NOTE : You should remember some of the Pythagorean triplets (e.g. 3,4,5 because \(5² = 3² + 4²\)). Some others are (5, 12, 13), (7, 24, 25) etc.