Triangles

 

When it comes to geometry, triangles become the most favorite topic of standardized exams and GRE is no exception. Triangles is a very important topic in the GRE Quantitative Reasoning section. Therefore it is crucial to understand its concepts and properties.

So what is a triangle and what are its properties?

Triangle is the simplest form of a polygon and is made up of just three sides. It is essential to remember that the sum of all the internal angles of a triangle is equal to 180 degrees.

Types of Triangles

Triangles can be categorized based on different attributes. Let us look at some of the attributes on which we can divide the triangles.

Based on Side Length

    1. Equilateral Triangle: All sides, as well as all angles of an equilateral triangle, are equal. Hence, \(Area \; of \; Equilateral \; Triangle = \frac{\sqrt{3}}{4}a^{2}\), where ‘a’ is the length of the side of equilateral triangle. Each angle of an equilateral triangle is equal to 60°.

Equilateral Triangle

    1. Isosceles Triangle: Two of the sides, as well as two angles of an isosceles triangle, are equal.

Isosceles Triangle

    1. Scalene Triangle: Neither any side nor any angles are equal in a scalene triangle.

Scalene Triangle

Based on Angles

    1. Acute Triangle: All the three angles in an acute angled triangle is less than 90°.

Acute Triangle

    1. Right Angle Triangle: If one of the angles of the triangle is equal to 90 degrees, such a triangle is known as right angled triangle.

Right Angle Triangle

    1. Obtuse Triangle: In an obtuse angled triangle, one of the angles in the triangle is greater than 90°.

Obtuse Triangle

Based on Length & Angle

    1. Right Isosceles Triangle: It is a combination of both isosceles as well as right angled triangle. So it holds all the properties of both isosceles as well as right angled triangle.

Right Isosceles Triangle

    1. Obtuse Isosceles Triangle: It is a combination of both isosceles as well as obtuse angled triangle. Hence, it holds all the properties of both isosceles as well as obtuse angled triangle.

Obtuse Isosceles Triangle

Properties of Similar Triangles

  • The corresponding angles of similar triangles are the same.
  • The corresponding sides of similar triangles are in the same proportion.
  • The corresponding medians of similar triangles are all in the same ratio.
  • The corresponding altitudes of similar triangles are all in the same ratio.
  • The ratio of the areas is the square of the ratio of the sides of similar triangle. For example, if the side lengths of triangle A is 4, 5, and 6 inches, and the side length of triangle B is 8, 10, and 12 inches, then the ratio of the sides of these triangles is 2:1, hence, the ratio of their areas will be \((2)^{2}:(1)^{2} = 4:1\)

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