EQUILATERAL TRIANGLE: CHARACTERISTICS, PROPERTIES, FORMULAS, AREA - MATHS - 2024

An equilateral triangle is a polygon with three sides, where they are all equal; that is, they have the same measure. For this characteristic it was given the name of equilateral (equal sides).

Triangles are polygons considered the simplest in geometry, because they are made up of three sides, three angles, and three vertices. In the case of the equilateral triangle, since it has equal sides, it implies that its three angles will also be.

An example of an equilateral triangle

Characteristics of equilateral triangles

- equal sides

Equilateral triangles are flat and closed figures, made up of three line segments. Triangles are classified by their characteristics, in relation to their sides and angles; the equilateral was classified using the measure of its sides as a parameter, since these are exactly the same, that is, they are congruent.

The equilateral triangle is a particular case of the isosceles triangle because two of its sides are congruent. So all equilateral triangles are also isosceles, but not all isosceles triangles will be equilateral.

In this way, equilateral triangles have the same properties as an isosceles triangle.

Equilateral triangles can also be classified by the amplitude of their interior angles as an equilateral acute triangle, which has three sides and three interior angles with the same measure. The angles will be acute, ie be less than 90 ^or.

- Components

Triangles in general have several lines and points that compose it. They are used to calculate the area, the sides, the angles, the median, the bisector, the bisector and the height.

• The median: it is a line that starts from the midpoint of one side and reaches the opposite vertex. The three medians meet at a point called the barycenter or centroid.
• The bisector: it is a ray that divides the angle of the vertices into two angles of equal measure, that is why it is known as the axis of symmetry. The equilateral triangle has three axes of symmetry. In the equilateral triangle, the bisector is drawn from the vertex of an angle to its opposite side, cutting it at its midpoint. These meet at a point called incenter.
• The bisector: it is a perpendicular segment to the side of the triangle that has its origin in the middle of it. There are three mediatices in a triangle and they meet at a point called the circumcenter.
• The height: it is the line that goes from the vertex to the side that is opposite and also this line is perpendicular to that side. All triangles have three heights that coincide at a point called the orthocenter.

In the following graph we see a scalene triangle where some of the components mentioned are detailed

The bisector, median and bisector are coincident

The bisector divides the side of a triangle into two parts. In equilateral triangles that side will be divided into two exactly equal parts, that is, the triangle will be divided into two congruent right triangles.

Thus, the bisector drawn from any angle of an equilateral triangle coincides with the median and the bisector of the side opposite that angle.

Example:

The following figure shows triangle ABC with a midpoint D that divides one of its sides into two segments AD and BD.

By drawing a line from point D to the opposite vertex, the median CD is obtained by definition, which is relative to vertex C and side AB.

Since segment CD divides triangle ABC into two equal triangles CDB and CDA, it means that the congruence case will be held: side, angle, side and therefore CD will also be the bisector of BCD.

A plotting segment CD, the angle of the vertex is divided into two equal angles of 30 ^or the angle of the vertex A still measuring 60 ^or and the line CD at an angle of 90 ^or with respect to the midpoint D.

The segment CD forms angles that have the same measure for the triangles ADC and BDC, that is, they are supplementary in such a way that the measure of each one will be:

Med. (ADB) + Med. (ADC) = 180 ^or

2 _* Med. (ADC) = 180 ^or

Med. (ADC) = 180 ^or ÷ 2

Med. (ADC) = 90 ^o.

And so, we have that segment CD is also the bisector of side AB.

The bisector and the height are coincident

By drawing the bisector from the vertex of one angle to the midpoint of the opposite side, it divides the equilateral triangle into two congruent triangles.

So that an angle 90 is formed ^or (straight). This indicates that that line segment is totally perpendicular to that side, and by definition that line would be the height.

Thus, the bisector of any angle of an equilateral triangle coincides with the height relative to the opposite side of that angle.

Ortocenter, barycenter, incenter, and coincident circumcenter

As the height, median, bisector and bisector are represented by the same segment at the same time, in an equilateral triangle the meeting points of these segments -the orthocenter, bisector, incenter and circumcenter- will be found at the same point:

Properties

The main property of equilateral triangles is that they will always be isosceles triangles, since isosceles are formed by two congruent sides and equilateral by three.

In this way, the equilateral triangles inherited all the properties of the isosceles triangle:

Internal angles

The sum of the angles is always equal to 180 ^or, as all angles are congruent, then each of these will measure 60 ^or.

External angles

The sum of the exterior angles 360 will always equal ^or therefore each external angle will measure 120 ^or. This is because the internal and external angles are supplementary, that is, when adding them they will always be equal to 180 ^o.

Sum of the sides

The sum of the measures of two sides must always be greater than the measure of the third side, that is, a + b> c, where a, b and c are the measures of each side.

Congruent sides

Equilateral triangles have all three sides with the same measure or length; that is, they are congruent. Therefore, in the previous item we have that a = b = c.

Congruent angles

Equilateral triangles are also known as equiangular triangles, because their three interior angles are congruent with each other. This is because all its sides also have the same measurement.

How to calculate the perimeter?

The perimeter of a polygon is calculated by adding the sides. As in this case the equilateral triangle has all its sides with the same measure, its perimeter is calculated with the following formula:

P = 3 _* side.

How to calculate the height?

Since the height is the line perpendicular to the base, it divides it into two equal parts by extending to the opposite vertex. Thus two equal right triangles are formed.

The height (h) represents the opposite leg (a), the middle of the side AC to the adjacent leg (b) and the side BC represents the hypotenuse (c).

Using the Pythagorean theorem, the value of the height can be determined:

3 _* l = 450 m.

P = 3 _* l

P = 3 _* 71.6 m

P = 214.8 m.

References

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