Properties of Triangle - Infinity Learn by Sri Chaitanya

A triangle is a polygon, it has three sides, three-angle and three vertices. triangle can be classified into the different type of triangle on the side and angles. we use properties of triangle to study detail about the triangle, but we also use them to compare two or more triangle. with the help of properties of triangle, we can determine equality and inequality of the triangle. let we will know about the properties of triangle.

Properties of triangle depend upon:

Learn about the properties of triangles, we need to know about the types of triangles on which triangles’ properties depend. some properties are based on their side and angles and and some properties of triangle are common.

Different types of triangles:

Types of triangle triangles are based on sides and angles.

On the basis of side of triangle, there are three type of triangle:-

Equilateral triangle:- if all the sides of a triangle are equal are called equilateral triangle.
Isosceles triangle:- if two sides of a triangle are equal are called isosceles triangle.
Scalene triangle:- if no any side of a triangle are equal are called scalene triangle.

On the basis of angle of triangle, there are three type of triangle:-

Acute angled triangle:- if the angle of an triangle are less than 90° are called acute angle triangle.
Right-angled triangle:- if one angle of a triangle is 90° are called right-angled triangle.
Obtuse angled triangle:- if one angle of a triangle is greater than 90° are called obtuse angle triangle.

Triangle and its properties:-

with the help of properties we can identify the relationship between sides and angles of a triangles.

some important properties if triangle

Angle sum property

If the sum of three interior angles of triangle is 180°.

In the above figure, Let ΔPQR is a triangle.

then, ∠P+ ∠Q + ∠R = 180°

Triangle inequality property

If the sum of the length of two sides of a triangle is greater than third side.

In the above figure we can see that ΔABC which show triangle inequality property.

If a = 5 unit, b = 7 unit,, c = 4 units, let we will will verify triangle inequality property as follow.

a + b > c ⇒ 5 + 7 > 4
c + a > b ⇒ 4 + 5 > 7
c + b > a ⇒ 4 + 7 > 5

Pythagoras Property :-

In the Pythagoras theorem, if in a right-angled triangle the square of the is equal to the sum of the square of other two sides.

we can write as

Hypotenuse² = Base² + Altitude².

or H² = B² + A²

Side opposite the greater Angle is the longest side:-

If the side opposite the greater angle is longest side. let us consider a triangle ΔABC, in which ∠B is the greatest angle. So. AC is the longest side.

Exterior Angle property:-

If the sum of the two opposite angle of triangle is equal to the exterior angle which is formed by extend the side of triangle.

Let ΔABC be triangle in which ∠E be the exterior angle.

So we can express such as

∠e = ∠a + ∠b

Congruence Property:-

In the congruence property, two triangle are congruent if their corresponding sides and angles are equal.

they are 4 condition for two congruent triangle.

SSS – property:-If all the three corresponding side of two triangle are equal .
SAS – property:- if two corresponding side and one angle are equal of two triangle.
RHS – property:- if one right angle, one hypotenuse and one side are common in two triangle.
ASA – property:- if two corresponding angle and one side are equal in two triangle.

ΔXYZ ≅ DEFΔ because its hold all the congruency condition.

∠XYZ = ∠DEF

∠YXZ = ∠EDF

∠YZX = ∠EFD

XY = DE

XZ = DF

YZ = EF

Some basic triangle properties depend on area and perimeter of triangle:-

Area of a triangle:- total space enclose by three sides of a triangle in 2-dimensional plane. mathematically we can express we can express such as.

A = ½ × b × h; where A is the area of triangle, ‘b’ is the base of the triangle and ‘h’ is the height of the triangle. unit of the area of triangle is square unit.

Perimeter:- The total distance of the boundary of a triangle is called the perimeter of the triangle. or we can say that the sum of all the side of a triangle is the perimeter of the triangle.
Heron’s formula:- Heron’s formula is used when we have to given that the sides of the triangle. we can find are without the height of the triangle on using the the Heron’s formula.

For finding the area of triangle, firstly we have to need the calculate the semi-perimeter(s)

we can express semi-perimeter as; S = a+b+c/2 .

Area of the triangle(A) = √s(s – a) (s – b) (s – c).

Height and distance of a triangle

Definition of Height and Distance:

height can be defined as the measurement of an object in the vertical direction. Distance can be defined as measurement of an object in horizontal direction.

Method of finding the height and distance:-

With the help of area of triangle:- we know that area of triangle is A = ½ × b × h; where A is the area of triangle, ‘b’ is the base of the triangle or the distance and ‘h’ is the height of the triangle.

If we have to given area and base of a triangle ,then

h = 2A ÷ b

If we have to given area and height of a triangle,then

b = 2A ÷ h

2. By Pythagoras Theorem:- Pythagoras theorem is used in heights and distance to find the longest distance. it also define as the sum of the square of two side is equal to the square of third side.

i.e.; Hypotenuse² = Base² + Altitude²

or Altitude² = Hypotenuse² – Base²

or Base² = Hypotenuse² – Altitude²

3. Using trigonometry function:- Trigonometry is the study of relationship between sides angles of a triangle with the help of trigonometry function we can find the relationship between height and distance.

Many terms are associated with height and distance.

Line of sight:- that imaginary line which can be drawn from the observer’s eye to an object is called the line of sight. the line of sight tells us about the types of inclination angle.
Angle of Elevation:- that angle which is formed by the line of sight with the horizontal when the viewer is viewing the object upward is called angle of elevation.
Angle of Depression:- that angle which is formed by line of sight with the horizontal when the viewer is viewing the object downwards. we can say that it is formed when the observer lower his head downward from top to the ground.

as shown in fig. line of sight, Angle of elevation and Angle of depression.

Trigonometric Ratios:-

we can find height and distance relationship between and problems are solved by using using trigonometry ratio. we have 6 trigonometric ratio.

sinθ, cosθ, tanθ, cosecθ, secθ, cotθ

Let us consider a right-angled triangle

Some specific Ratios of Trigonometry:-

Some standard value of angles 0°, 30°, 45°, 60°, and 90° on we can find the value of height and distance.

let we do define a table for trigonometry ratios sine, cosine, cotangent, tangent, secant, and cosecant for above angles

Example:- Let an observer height is 1.5m is 28.5m away from a tree. The angle of elevation of the top of the tree is measuring 45. what is the height of the tree?

Solution:-

from the above fig.

BD = CE = 1.5m

In ΔABC,

∠B = 90°

⇒tanθ = P/B = AB/BC

⇒tan45° = AB/28.5

⇒ 1 = AB/28.5

⇒ AB = 28.5

Now, h = AB+BD = 28.5+1.5 = 30

So, the height of the tree is 30m

FAQ’s

What is the perimeter of a triangle

sum of all the sides of a triangle. Let ABC is a triangle in this triangle AB, BC and CA are side of triangle. then, perimeter of the triangle = AB + BC + CA

What is the formula of heron's formula ?

Semi-perimeter as (S) = a+b+c/2 . Area of the triangle(A) = √s(s - a) (s - b) (s - c)

which type of triangle we used in the problem of distance and height ?

Most commonly triangle is used to solve the problem of distance and height is a right-angled triangle.

Q. Classify the types of triangle.

Ans. We can classify the triangle into two categories

On the basis of side of triangle, there are three type of triangle:-

Equilateral triangle:- if all the sides of a triangle are equal are called equilateral triangle
Isosceles triangle:- if two sides of a triangle are equal are called isosceles triangle.
Scalene triangle:- if no any side of a triangle are equal is called a scalene triangle

On the basis of angle of triangle, there are three type of triangle:-

Acute angled triangle:- if the angle of an triangle are less than 90° are called acute angle triangle.
Right-angled triangle:- if one angle of a triangle is 90° are called a right-angled triangle.
Obtuse angled triangle:- if one angle of a triangle is greater than 90° are called obtuse angle triangle.