PROPERTIES OF TRIANGLE RADII

 Triangle was discovered by a Chinese Mathematician Jia Xian in the 11th century and it came to light when it was again studied by Yang Hui who is also a Chinese mathematician in the 13th century. Due to the invention of the triangle, China is also called Yanghui Triangle. The Triangle name has been taken from the Latin word ‘triangulus’ which means having three edges or three angles. 

The circle which is found inside the triangle is not a usual circle, it is a special circle as it has 3 edges of a triangle which are its tangents. It is the biggest circle present in the triangle and at the centre, it is at the cross-section of the bisector of the 3 angles which are there in a triangle. 

FORMULA FOR CALCULATING THE RADIUS OF A TRIANGLE:

Triangle radii mean the inradius which is generally written by the letter r. So the formula is :

r = K/s. where k refers to as the area of the triangle and s refers to as the semi perimeter. Now we have to assume the bases triangle as ABC and with a height ( r ) each. Now it is said that the line joining the incentre should break down the triangle into 3 parts, which will lead to the area of the triangle as a r / 2 + b r / 2 + c r / 2.

we can also apply Heron’s formula which is:

r =√ (s-a)(s-b)(s-c)/s . 

there is another formula that can be applied for finding out radius is:

r = 4 R sin(a / 2 ) sin ( b / 2 ) sin ( c / 2 ).

Where a, b, c refers to the sides of the triangles.

AREA OF A TRIANGLE:

The area of a triangle means the total space that is taken by the 3 corners of the triangle. it is generally half of the size of the base height. So the formula for calculating the area of a triangle is:

A = ½ *b*h. 

Here, A refers to as the area and b refers to as the base and h refers to the height of the triangle. It is termed in square units. 

Calculation of the area of a triangle is:

Example:

Find out the area of a triangle with a base of 4 cm height of 6 cm.

Solution. 

b = 4 cm, h = 6 cm 

therefore, area of triangle’s formula is:

A = ½ *b*h 

A = ½ * 4*6 = 12 cm².

PROPERTIES OF TRIANGLES:

Properties of triangle radii have many statements which are proven too. Some of the statements are as follows:

STATEMENT: In ΔABC, r1=Δ s − a.

PROVE:

Let I₁ be the ex centre of ΔABC opposite to the vertex A.

Properties of Triangles – EX-RADII

Let X, Y, Z be the projections of I₁ on BC, CA, AB. ∴ I₁X = I₁Y = I₁Z = r₁

Δ = Area of ΔABC

= Area of Δ AI₁ B + Area of ΔAI₁ C – Area of ΔBI₁ C

= ½ AB.I₁X + ½AC.I₁Y – ½BC.I₁Z = ½c r₁ + ½b r₁ – ½a r₁

= ½r₁ (c + b – a) = ½r₁ 2(s – a) = r₁ (s – a) → r₁ = Δ / s – a.

Similarly, r₂ = Δ / s – b, r₃ = Δ / s – c.

STATEMENT: In ΔABC, r₁ = 4R sin A / 2 cos B/2 cos C / 2.

PROVE:

4 R sin A / 2 cos B/2 cos C / 2 = 4R s-bs-cbc  * ss-bca * ss-cab

= 4 R s ( s – b ) ( s – c ) / a b c = 4 R s ( s – a ) ( s – b ) ( s – c ) /  ( s – a )4 R Δ 

= Δ² / ( s – a ) Δ = Δ/ (s – a) = r1. 

CONCLUSION:

The main objective is to know the various types of angles there and to evaluate the angles present there in the circle using triangle properties and circle properties. Hypotenuse refers to the diameter of the circumscribed circle in the case of a right-angle triangle, which means the center is the midpoint of the hypotenuse. Centre of the triangle is referred to as the point of intersection of the perpendicular sides of a triangle. The centre of the inscribed circle is the cross-section area of the bisector of the angles in any triangle.