Basic properties of triangles
- Sum of all interior angles of a triangle = 180°
- Largest angle is opposite to the longest side in a triangle or the longest side is opposite to the largest angle.
- Smallest angle is opposite to the shortest side or the shortest side is opposite to the smallest angle.
i.e. if x° > y° > z° then a > b > c.
- Angles opposite to the equal sides are equal and the sides opposite to the equal angles are equal.
- Sum of any two sides is greater than the third side.
- Difference of any two sides is lesser than the third side.
- Exterior angles of a triangle.
|Exterior angle theorem:
An exterior angle of a triangle is equal to the sum of its opposite interior angles.
∠PRS = ∠P + ∠Q
- Sum of all exterior angles = 360°
Classification of triangles as per angles
- Acute angled triangle:
In a triangle if all angles are less than 90° is called an acute angle. In an acute angled triangle the square of the side opposite to the acute angle is less than the sum of the squares of the other two sides.
- Right angled triangle:
In a triangle if exactly one angle is equal to 90° and rest two are acute, then it is a right angled triangle. In a right angled triangle the square of the side opposite to right angle is equal to the sum of the squares of the other two sides. This result is known as Pythagoras theorem and we will discuss it in detail later.
- Obtuse angled triangle:
In a triangle if exactly one angle is obtuse and rest two angles are acute, then it is an obtuse angled triangle. In an obtuse angled triangle, the square of the side opposite to the obtuse angle is greater than the sum of the squares of the other two angles.
Classification of triangles as per sides
- Equilateral triangle:
If all the three sides of a triangle are equal in length, then the triangle is equilateral.
Each angle of an equilateral triangle = 60°.
- Isosceles triangle:
Any two sides of a triangle are equal, then the triangle is isosceles.
In diagram sides FG = FH, so ∠G = ∠H.
∠G and ∠H are called the 'base angles' and ∠F is called the vertical angle.
- Scalene triangle:
If all the three sides of a triangle are different in length then the triangle is scalene.
Geometrical centers of a triangle
- Circum center(S)
The point of concurrency of the perpendicular bisectors of the three sides of a triangle is called the Circum center of the triangle.
In the diagram, S is the circum center and SA = SB = SC = R, where R is the circum radius.
In an acute angled triangle circum center lies inside the triangle. In a right angled triangle, circum centre is the midpoint of the hypotenuse, hence the circum radius = 1/2 (hypotenuse). In an obtuse angled triangle circum centre lies outside the triangle.
- In centre of a triangle (I)
In center is the point of concurrency of all the angle bisectors of a triangle.
In the diagram, 'I' is the in centre and ID is a perpendicular drawn to the side BC.
ID is the in-radius of the triangle and that is denoted by 'r'.
∠BIC = 90° + 1/2 ∠A
AIC = 90° + 1/2 ∠B
AIB = 90° + 1/2 ∠C
Point of concurrency of the internal bisector of any one angle and the external bisectors of the other two angles is called 'Ex-center' of the triangle.
- Ortho center of a triangle (O)
Point of concurrency of the three altitudes of a triangle is called the ortho-center and it is denoted by 'O'.
In an acute angled triangle ortho-center lies inside the triangle.
In a right angled triangle ortho center is the vertex where right angle formed.
In an obtuse angled triangle ortho center lies outside the triangle.
AOC = 180° - ∠B
BOC = 180° - ∠A
AOB = 180° - ∠C
- Centroid of a triangle (G)
Centroid is the point of concurrency of all the medians.
The line segment joining one vertex and the midpoint of its opposite side is called median.
D, E and F are the mid points of the sides BC, AC and AB respectively.
AD, BE and CF are the medians.
G is the Centroid.
- In a right angled triangle the length of the median drawn to the hypotenuse is equal to half the length of the hypotenuse.
- Centroid divides medians in the ratio 2:1.
i.e. Ag/DG = BG/EG = CG/FG = 2/1
Median divides the triangle into two equal areas.
i.e. Area of ABE = Area of ΔCBE = Area of ΔACF = Area of ΔBCF = Area of ΔABD = Area of ΔACD = 1/2 & Area of ΔABD
Similarity and congruency of triangles
When two angles are similar in shape, then the triangles are similar triangles.
If two triangles are similar, then their sides are in proportion.
Then; AB/PQ = BC/QR = AC/PR = Perimeter of ΔABC/Perimeter of ΔPPerimeter of ΔPQR = √Area of ΔABC/√Area of ΔPQR
When two triangles are equal in size and shape, then they are congruent.
ΔABC ≅ ΔDEF
AB = DE, BC = EF and AC = DF
Perimeter of ΔABC = Perimeter of DEF
Area of ΔABC = Area of ΔDEF