9 junio, 2024

Types of triangles and their characteristics (with examples)

A triangle is a polygon or geometric figure that has three sides, three vertices, and three angles. The sides are each of the straight lines that form it. The vertices are the points where the sides meet; the angles are the arcs or openings that are formed near the vertices, when joining two sides.

A triangle can also be defined as the area determined by three lines. The sum of its three angles is always equal to 180º. The length of any of its sides is always less than the sum of the lengths of the other two sides, but greater than their subtraction.

Triangles are the simplest geometric figures, and they are used to investigate the mathematical properties of other more complex figures, such as pentagons or hexagons.

They are also used in other sciences, such as topography, navigation or astronomy. In the latter, they are used to determine the distance that separates us from a distant celestial body from two observation points located on Earth. This method is known as parallax.

Triangles are classified according to the length of their sides or the width of their angles.

Types of triangles according to their sides

Equilateral triangle

The sides of this type of triangle are exactly the same length. And the same happens with its angles: all three measure 60º. That is why we say that the equilateral triangle is a regular polygon.

Scalene triangle

Unlike the equilateral, in the scalene triangle everything is unequal: its three sides have different lengths and their angles differ in width.

Isosceles triangle

In this type of triangle we find that two sides have the same measure, while the remaining side is different. The same is observed in the amplitude of the angles: two are equal and one is different.

Types of triangles according to their angles

Right triangle

It is characterized by having a right angle, that is, 90º. Its other two angles are acute or less than 90º.

In this type of triangle, the longest side is called the hypotenuse, while the other two sides are the legs.

oblique triangle

Triangles that do not have any right angles belong to this type. They are subdivided into two types:

acute triangle: its three angles are acute.

Obtuse triangle: they have two acute angles and one obtuse or greater than 90º.

mixed triangles

The same triangle can be classified according to the two criteria, that is, according to the length of its sides and the width of its angles.

For example, a right triangle can also be scalene or isosceles, but it could not be equilateral, since the latter does not present any right angle.

However, an equilateral triangle could be acute, since it effectively has three acute angles or angles less than 90º.

A scalene triangle can be at the same time obtuse, since both the amplitude of its angles and the length of its sides are different.

How to calculate the perimeter of a triangle?

The product of the sum of the lengths of the three sides of a triangle is called the perimeter.

Let’s see some examples.

1- We are asked to find the perimeter of a scalene triangle whose sides are 6, 8 and 4 centimeters. All we have to do is add:

6 + 8 + 4 = 18

Therefore, the perimeter of this scalene triangle is 10 centimeters.

2- Next, they ask us to calculate the perimeter of an isosceles triangle whose sides measure 4 centimeters, the two equal ones, and the remaining side 6 centimeters. Since two of its sides have the same length, we must place the same number twice, like this:

4 + 4 + 6 = 14

The perimeter of this triangle is 14 centimeters.

3- One last example. We have the task of determining the perimeter of an equilateral triangle with sides 9 centimeters. Since we already know the characteristics of the various types of triangles, we know that the equilateral is distinguished because its three sides are equal. Therefore:

9 + 9 + 9 = 27

The perimeter of this equilateral is 27 centimeters.

bisectors, bisectors and medians

These are the three types of straight lines that can be drawn in a triangle.

bisectors

There are three, one for each side of the triangle. The perpendicular bisector is a straight line that passes through the midpoint of the side of the triangle to which it corresponds. The three perpendicular bisectors of a triangle intersect at a point known as the circumcenter, which is the same distance from each of the vertices of the triangle.

Bisectors

There are three, one for each angle. The bisector is a straight line that starts from the vertex and divides the angle into two equal parts. The bisectors of a triangle intersect at a point known as the incenter.

medians

There are also three, one for each vertex. A median is a line that starts at a vertex and goes to the midpoint of the opposite side. The medians of a triangle intersect at a point called the centroid.

The distance between any of the three vertices and the center of gravity is equal to two thirds (2/3) of the total length of the corresponding median. For example, if the median CE measures 5 centimeters, then the distance between C and the centroid (O) is equal to 5 x 2/3, or what is equal, to 5 x 0.66, which results in 3, 3 centimeters.

heights

It is a straight line that joins a vertex with the opposite side. The three altitudes of a triangle intersect at a point called the orthocenter. Depending on the type of triangle, the orthocenter can be inside or outside the area of ​​the triangle.

How to calculate the area of ​​the triangle?

The area of ​​a triangle of any type can be found by applying the following formula:

A = b x h / 2

In this equation, A refers to area; b refers to the base and h is the height.

Let’s see an example. We are asked to calculate the area of ​​a triangle whose base is 12 centimeters and whose height is 7 centimeters. Thus, we have:

b = 12

h = 7

We apply the formula:

A = 12 x 7 / 2

A = 84 / 2

A = 44

This triangle therefore has an area of ​​44 square centimeters.

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