Properties and Solutions of Triangle

Triangle is one of the simplest shapes. However, it does not mean that they are not necessary! Nevertheless, triangles are not just mathematically significant. Triangles and its properties are unique because they are powerful. We can make straight underpinnings of metal out of all the two-dimensional shapes. A triangle has a rigid shape, whereas all other shapes can be distorted with a slight push that exists hinged at the corners. Triangles are also unique because they are the simplest polygon — a common approach to a tricky geometrical problem, such as analysing a complex surface, is to approximate it by a mesh of triangles.

Triangles 

A triangle is a three-dimensional polygon that consists of three vertices and three edges. The important property of a triangle is that the sum of the internal angles of a triangle is equal to 180 degrees.

Triangles and its properties

The triangle and its properties help us quickly identify a triangle from a given set of figures. They are divided into

Triangles and it’s primary properties

A triangle consists of 3 angles and 3 sides.

Degrees: The aggregate angles of the triangles are always 180 degrees. The external angles of the triangles are always 360 degrees.

Angles: The sum of successive exterior and interior angles is supplementary.

Length: The sum of the two sides of a triangle is always greater than the third. Likewise, the difference between any two sides of a triangle is less than the third side.

Sides: The triangle’s longest side is always opposite the largest interior angle. In comparison, the triangle’s shortest side is opposite the shortest exterior angle.

Triangles and its properties related to Perimeter and Area

Perimeter: The perimeter of a triangle = the sum of all the triangle sides.

Area: It is measured in square units. The area of a triangle’s formula is Area (A) = (1/2) × Base × Height.

Heron’s formula: It is also one of the properties used to calculate the area of a triangle. If a triangle with sides c, d, and e, the area is given by; A = √s(s−c)(s−d)(s−e) and the semi-perimeter (s) = (c + d + e)2

Circles Connected with Triangles.

1. Circumcircle and radius of a triangle

•  Circumcircle: The circle of a triangle is defined as the angular points that pass through the triangle, called the circumcircle. The point of intersection of a perpendicular bisector is the circle’s centre, called the circumcentre. R. denotes the radius. The circumcentre may lie outside, within, or upon the sides of the triangle.

2. In-circle or Inscribed circle of a Triangle

• Inscribed / In-circle: The circle inscribed within the triangle to connect each side of a triangle is called an in-circle or inscribed circle. Its radius is denoted by r, which is equal to the length of the perpendicular from its centre to any of the sides of the triangle.

3. Escribed circle of a triangle 

• Escribed circle:  The escribed triangle is the circle that touches the sides and produces a triangle. Also, the internal bisector passes through the same point. Its radius is denoted by r. The centre of the described circles is known as ex-centres. The centres of the described circles are called the ex-centres. 

4. Centroid: The centre point in which the three medians intersect is called the       Centroid of the triangle. The Centroid is the point of intersection of all the medians of the triangle.

5. Orthocentre: An orthocenter is the point of intersection of lengths drawn perpendicular to the opposite sides. The three main elements of an orthocenter are

1. Triangle
2. Altitude
3. Vertex

Solution of triangles

The Solution of triangles is one of the fundamental trigonometric problems that find Solutions for the characteristics of the triangles based on the lengths, sides, and angles. The Solution of triangles can be divided into two divisions

1. Solution for plane triangles
2. Solution for spherical triangles.

The solution of Plane triangles

is based on six characteristics, whereas three of the six characteristics determine the other three.

1. SSS- Includes all the three sides
2. SAS – Two sides and one angle.
3. SSA – Two sides and an angle not included if the side length of the adjacent angle is shorter than the other side.
4. ASA – Two adjacent angles and one side

Solution for spherical triangles

1. AAA – Includes three angles.
2. SAS – Two sides which include an angle
3. SSS- Includes three sides
4. SSA – Two sides that include a non-angle
5. ASA – Two adjacent angles and aside.
6. ASA – An adjacent angle, one side, and opposite angle.

Conclusion

Triangles are always special because they are the easiest polygon — a familiar path to a problematic geometrical issue. This approach is also employed worldwide to accomplish some of the unfamiliar shapes we now see in modern architecture. The triangulation method is essential in creating our virtual world. The CGI qualities we see in movies and television are usually matched by a beautiful mesh of triangles, making it easier to store and exploit them digitally. Triangles are the uncomplicated shapes that make our mathematical, physical, and digital worlds round.