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Introduction
The concept of similar triangles has various uses in our daily lives. However, to analyze those areas it is significant to understand the theory of similar triangles. In a further study, similar triangles have been defined from several views of mathematics.
There are some particular properties of similar triangles that help two triangles to be congruent and correspond to each other. Similar triangles have many different application areas as such with the help of the properties of similar triangles the heights of any object, length of any tower or building can be calculated. In the discussion, these properties have been mentioned.
Discussion
What is a similar triangle?
In mathematics, similar triangles refer to two triangles where the angles of each triangle are corresponding to each other. In mathematics, similar triangles are also identical in form in terms of their figures. Each side of the triangles which are corresponding to each other is also considered in comparative relation as a whole.
It can be also said that similar triangles are not different in shape, however, in the geometric calculation, two similar triangles are not necessarily the same in size. According to mathematics, similar triangles are harmonious in a condition that the equivalent sides of the similar triangles are the same in length.
The length of each side of the triangle is constant in its ratio compared to the length ratio of the other triangle. This similarity in the length ratio of two similar triangles is known as the “scalar factor” under mathematics terminology. The similarity of the two triangles is represented by the symbol ‘~’.
The mathematics representation of the scalar factor or scale factor is, △ ABC is a similar triangle of △ EFG then the representation under the mathematics will be AB/EF= AC/EG= BC/FG.
Therefore for example △ ABC ~ △ EFG, and suppose AB= 3, BC= 4, AC = 5, and EF = 12 has been mentioned then by applying the condition of similar triangles it can be calculated the value of EG and FG.
As already mentioned in the equation where AB/EF= AC/EG= BC/FG, therefore it can be said that 3/12 = 5/ EG = 4/ FG. Since each side of the triangles are corresponding to each other.
Then as per the calculation, the scalar factor will be 3/12 = ¼. Hence as per the equation, the value of the EG side of the second triangle is 20 and the value of the FG side is 16.
Characteristics of similar triangles
As mentioned before two triangles will be called similar if the triangles are also the same in their shapes. Hence the characteristics of the similar triangle are defined through two aspects, these are three same angles and three same sides. These are termed “AAA” and “SSS” under mathematics geometry calculation. These are discussed below:
“AAA” (“angle angle angle”): In this aspect or condition of similarity all three angles of the triangle that is analogues with the second triangle are represented as the “similar triangle AAA”.
“SSS” (“side side side”): From this aspect two triangles will be called similar to each other if each side of the first triangle is equivalent to the second triangle. Therefore the two triangles will be denoted as similar triangles in the condition of the same sides.
Apart from these two conditions, two triangles will be considered to be corresponding to each other if they meet the condition of two similar sides and one proportional angle. Under mathematics, this condition is known as “SAS” (“side angle side”), where two triangles are determined with two equivalent sides in terms of length and one angle that is common in both the triangles.
Properties of similar triangle
Under mathematics properties of similar triangles are important to understand as the properties of similar triangles help to evaluate various geometric calculations. These properties of similar triangles are also helpful for practical application in architectural studies. Apart from this mathematical physics is applied to determine various laws. Some properties of a similar triangle are mentioned below:
Each angle of the triangle that is corresponding is also identical to each other, therefore A = E, B = F, and C = G from the above problem.
Secondly, the similar sides that are corresponding with the sides of the second triangle are also having the same dimension. Therefore in mathematics, this can be shown as AB/EF= AC/EG= BC/FG.
Other than that under the properties of similar triangles one more aspect is two similar angles and one similar side. As per this condition, two triangles will be considered as a similar triangle if any two angles of both triangles and anyone side are the same.
Theorems of similar triangle
As mentioned earlier any triangle is denoted with three sides and three angles. These sides and angles when enclosed, form a triangle. The total value of these angles is 180 degrees, and any trigonometric shape whose total value by adding the angles together is 180 degrees is called a triangle. This comes under the theory of triangles, furthermore, suppose a triangle ABC,
“∠ CAB + ∠ ABC + ∠ BCA = 180°” where three angles are equal to 90°, proposing the summation of 180°.
If ∠ B = 90°
∠A + ∠ C = 90°
Therefore, “∠A + ∠B + ∠C = 180°”.
Conclusion
Triangles and their related theories are significant in mathematics as the triangles are the steadiest figure which is decomposed by arbitrary polygons. Hence while studying mathematics it is important to have a clear knowledge of every concept of triangles. Similar triangles are the one common variety among all the triangles. In the discussion, the concept of similar triangles and properties of similar triangles has been elaborated for better understanding.
