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The Triangle is the most important topic in geometry, and it is covered in depth in numerous competitive tests such as the CAT. Numerous triangle-related questions are dependent on fundamental ideas like the understanding of sides and angles, as well as a few rudimentary triangle-related theorems.
It’s fascinating to observe that questions may be found on even the most basic of notions, which is unusual. In this article, we’ll go over two fundamental triangle principles and how they might be used in practical situations.
Sides of triangles
Types of triangles
The triangle
A triangle is a polygon with three sides and three vertices. A triangle is formed by connecting three non-collinear points; otherwise, it would be a straight line. The sum of the three interior angles is 180 degrees, while the sum of the outer angles is 360 degrees. Different varieties of triangles exist. They are categorized according to the dimensions of their sides and angles. Additionally, there is a comparison between two triangles, such as congruence, likeness, etc.
The sum of any two sides of a triangle is always greater than the length of the third side, i.e. AB + BC > AC and the difference between two sides is always less than the length of the third side, | AB – BC |< AC.
An exterior angle’s value equals the sum of two opposite internal angles.
The side opposite the largest internal angle is the triangle’s longest side and vice versa.
The Heron Formula:
The simplest way to calculate the area of a triangle is
Triangle area = 1/2 x base x height
When the height is unknown, but the lengths of all sides are known, there is a simple formula for calculating the area of a triangle. You do not need to calculate the height before determining the area.
Then, let a, b, and c represent the lengths of the triangle’s sides.
(s*(s-a)*(s-b)*(s-c))1/2
where s equals (a+b+c)/2
‘s’ is also known as the semi-perimeter because it represents half of the perimeter’s value.
Congruence and Similarity
Triangle congruence and resemblance is a key topic in CAT geometry fundamentals. Here are some fundamentals regarding the two notions. Several contenders mistake the two notions, necessitating a comparison.
Congruency
- Two triangles are said to be congruent if they have the same size and shape. All interior angles and their associated sides have identical dimensions.
- SAS rule – Two triangles are congruent if two sides of one triangle are identical to corresponding sides of another triangle and the angle between the two triangles is likewise equal.
- SSS rule – Two triangles are SSS congruent if the three sides of one triangle match the three sides of the other triangle.
- AAS rule- If two corresponding angles are equal and one corresponding side that is not included in the triangle is also equal in length, then the triangles are said to be AAS congruent.
- ASA rule- This rule only applies to congruency. It states that two triangles are congruent if the two corresponding angles and the side included between them are equal.
Similarity
- Two triangles are considered to be similar when all of their corresponding angles are the same and the lengths of their respective sides have the same ratio.
- SAS rule – When two corresponding sides in the two triangles are in the same proportion as each other and the corresponding angles in the two triangles are likewise equal, two triangles are said to be similar
- SSS rule- In this case, if the proportions of the three sides of the first triangle are identical to those of the three sides of the second triangle, the two triangles are said to be SSS similar.
- AA rule – If two pairs of angles in two different triangles are equal, two triangles are said to be AA similar.
- ASA rule – there is no ASA rule applicable for similarity.
Conclusion
In this article, we have covered two fundamental triangle principles and how they might be used in practical situations. A triangle is a polygon with three sides and three vertices. The sum of any two sides of a triangle is always greater than the length of the third side. Two triangles are considered to be similar when all of their corresponding angles are the same and the lengths of their respective sides have the same ratio.
