TRIANGLES





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 \(5^2 = 3^2 + 4^2\)). Some others are (5, 12, 13), (7, 24, 25) etc.




Isosceles Right Triangle




A right triangle whose two legs are equal is an isosceles right triangle. The ratio of sides is \(1 : 1 : \sqrt{2}\)


30, 60, 90 triangle



This is a special case of a right triangle whose angles are 30°, 60°, 90°.

In this triangle side opposite to angle \(30° = \frac{HYP}{2}\), Side opposite to Angle \(60° =\) \(\frac{\sqrt{3}}{2} \times HYP\). The ratio of sides is \(1 : \sqrt{3} : 2\)



Equilateral Triangle



A triangle whose all sides are equal is called an equilateral triangle. If a be the side of an equilateral triangle, then

1. Area \(= \frac{\sqrt{3}}{4} a^2\)

Altitude \(= h=\frac{\sqrt{3}}{2}a\)

2. Given the perimeter, equilateral triangle has the maximum area.

3. Of all the triangles that can be inscribed in a circle, the equilateral triangle has the greatest area.

4. Area of Outer Circle is 4 times the area of Inner Circle

5. The triangle formed by joining the mid-points of the sides will be half in perimeter and one-fourth in area

6. For similar figures and solids, \(\frac{A_1}{A_2} = \left( \begin{array}{cc} \frac{\ell_1}{\ell_2 } \end{array} \right)^2\) and for similar solids \(\frac{V_1}{V_2} = \left( \begin{array}{cc} \frac{\ell_1}{\ell_2 } \end{array} \right)^3\)

7. A triangle with 2 sides (and two angles) equal is called isosceles.



GENERAL THEOREMS ON SIMILARITY

Proportionality Theorem



Intercepts made by two transversal lines (cutting lines) on three or more parallel lines are proportional. In the figure, lines \(X_1, Y_1,\) & \(X_2Y_2\) are transversals cutting the three parallel lines AB,CD, EF. Then AC, CE, BD, DF are intercepts

Also, \(\frac{AC}{BD} = \frac{CE}{DF}\)


Midpoint Theorem

A triangle, the line joining the mid points of two sides is parallel to the third side and half of it.


Basic Proportionality Theorem




A line parallel to any one side of a triangle divides the other two sides proportionally. If DE is parallel to BC, then

(a) \(\frac{AD}{BD} = \frac{AE}{EC}\),
(b) \(\frac{AB}{AD} =\frac{ AC}{AE}\),
(c) \(\frac{AD}{DE} = \frac{AB}{BC}\) and so on.



PROPERTIES OF SIMILAR TRIANGLES

(1) Ratio of areas = Ratio of squares of corresponding sides
(2) RIGHT TRIANGLE :



ABC is a Right Triangle with A as the Right angle. AD is perpendicular to BC then

(a) Triangle \(ABD ~ Triangle CBA\) & \(BA^2 = BC \times BD\)
(b) Triangle \(ACD ~ Triangle BCA\) & \(CA^2 = CB \times CD\)
(c) Triangle \(ABD ~ Triangle CAD\) & \(DA^2 = DB \times DC\)



Some More Important Points:

1. If with a given perimeter, different figures are formed like equilateral triangle, square, regular hexagon, regular octagon ...... and eventually a circle (a regular polygon of infinite sides), then the triangle will have the minimum area and circle will have the maximum area.
2. If different triangles are inscribed in a circle, then the equilateral triangle will have the maximum area.
3. If the perimeter of a triangle is fixed, then the equilateral triangle will have the maximum area.
4. If the sum of two sides of a triangle is constant, then the isosceles right angled triangle will have the maximum area.