Equilateral triangle

🏆Practice types of triangles

Definition of equilateral triangle

The equilateral triangle is a triangle that all its sides have the same length.

This also implies that all its angles are equal, that is, each angle measures 60° 60° degrees (remember that the sum of the angles of a triangle is 180° 180° degrees and, therefore, these 180° 180° degrees are divided equally by the three angles).

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Test yourself on types of triangles!

Calculate the size of angle X given that the triangle is equilateral.

XXXAAABBBCCC

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Next, we will see some examples of equilateral triangles:

Equilateral triangle

A2 - image of equilateral triangle

Examples of equilateral triangles

A3-Examples of equilateral triangles


Another name for the equilateral triangle

Recall that a regular polygon is a geometric figure that:

  • Has all its sides equal
  • Has all its angles equal

Therefore, the equilateral triangle is also known as the regular three-sided polygon.

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Other characteristics of equilateral triangles

Recall that within a triangle there are what are called remarkable lines, which are the heights, medians, perpendicular bisectors and bisectors, these lines intersect at the so-called remarkable points (orthocenter, barycenter, circumcenter and incenter respectively).

  • In an equilateral triangle, the remarkable lines coincide.
  • In an equilateral triangle, the remarkable points coincide at the same point.

Recall that triangles can be classified according to the measure of their interior angles. Within this classification we find the acute triangles which are characterized by having all its acute angles (less than 90° 90° degrees).

Since an equilateral triangle has all its interior angles equal to 60° 60° degrees, it is also an acute triangle.


If you are interested in learning more about other triangle topics, you can enter one of the following articles:

On the Tutorela blog you will find a variety of articles about mathematics.


Equilateral Triangle Exercise

Exercise 1

image Given equilateral triangle ABC

Given the triangle ACB \triangle ACB equilateral

Task:

What is the value of angle ACB ∢ACB ?

Solution:

Given the equilateral triangle ACB ∆ ACB

In an equilateral triangle all its angles are 60° 60° .

Therefore the angle ACB ∢ACB is equal to 60° 60°

Answer:

60° 60°


Do you know what the answer is?

Exercise 2

Given the equilateral triangle:

Exercise 2 Given the equilateral triangle

Homework:

What is the perimeter?

Solution:

Since we are given an equilateral triangle, we will multiply the given side by 3 and get the perimeter of the triangle.

35=15 3\cdot 5=15

Answer: 35=15 3\cdot 5=15


Exercise 3

Given the equilateral triangle:

Exercise 3 Given the equilateral triangle

Task:

The perimeter of the triangle is equal to 33cm 33 cm . What is the value of X X ?

Solution:

Since we are given an equilateral triangle whose perimeter is 33cm 33 cm , all we have to do is divide the circumference by 3 and we get the side measure X X .

33:3=11 33:3=11

Answer: 1111


Check your understanding

Exercise 4

In the figure we are given an equilateral triangle.

The length of each side is equal to 7cm 7 cm

For each side there is a semicircle.

Exercise 4 In the figure we are given an equilateral triangle

Task:

What is the area of the whole figure? Replace a π=3.14 \pi=3.14

Solution:

S=S1+3S2 S=S1+3S_2

When S= S= the area of the whole figure

Area of the triangle S1= S_1=

Area of the semicircle S2= S_2=

In an equilateral triangle the height merges with the middle and so when ADBCAD⊥BC

3.5=127=DC=BD 3.5=\frac{1}{2}\cdot7=DC=BD

Right triangle: ∆ADC ∆\text{ADC}

We perform pythagoras:

AD2+DC2=AC2 AD²+DC²=AC²

AD2+3.52=72 AD²+3.5²=7²

AD2+DC2=AC2 AD²+DC²=AC²

AD2=732 AD²=\frac{7\sqrt{3}}{2}

S1=ADBC2=7327221.22cm2 S_1=\frac{AD\cdot BC}{2}=\frac{\frac{7\sqrt{3}}{2}\cdot7}{2}≈21.22\operatorname{cm}²

S2=12(Diaˊmetrodelcıˊrculo=7cm)=12(radio3.5cm) S_{2=}\frac{1}{2}(Diámetro del círculo=7 cm)=\frac{1}{2}(radio 3.5\operatorname{cm})

S2==12π3.52=123.143.52S_{2=}=\frac{1}{2}\cdot π\cdot 3.5²=\frac{1}{2}\cdot 3.14\cdot 3.5²

19.23 ≈19.23

S=21.22+319.23=78.91 S=21.22+3\cdot 19.23=78.91

Answer: 78.91 78.91


Ejemplos y ejercicios con soluciones de triángulo equilátero

Exercise #1

Is the triangle in the drawing an acute-angled triangle?

Video Solution

Step-by-Step Solution

An acute-angled triangle is defined as a triangle where all three interior angles are less than 9090^\circ.

In examining the visual depiction of the triangle provided in the problem, we need to see if it appears to satisfy this property. The assessment relies on observing the triangle's structure shown in the drawing and noting any geometric indications suggesting angle types.

Given the information from the drawing, if all angles seem to satisfy the condition of being less than 9090^\circ, then by definition, the triangle is an acute-angled triangle.

Conclusively, the answer to whether the triangle is acute-angled based on provided visual assessment and inherent assumptions in its illustration is: Yes.

Answer

Yes

Exercise #2

In an isosceles triangle, the angle between ? and ? is the "base angle".

Step-by-Step Solution

An isosceles triangle is one that has at least two sides of equal length. The angles opposite these two sides are known as the "base angles."
The side that is not equal to the other two is referred to as the "base" of the triangle. Thus, the "base angles" are the angles between each of the sides that are equal in length and the base.
Therefore, when we specify the angle in terms of its location or position, it is the angle between a "side" and the "base." This leads to the conclusion that the angle between the side and the base is the "base angle."

Therefore, the correct choice is Side, base.

Answer

Side, base.

Exercise #3

Given the values of the sides of a triangle, is it a triangle with different sides?

9.19.19.19.59.59.5AAABBBCCC9

Video Solution

Step-by-Step Solution

As is known, a scalene triangle is a triangle in which each side has a different length.

According to the given information, this is indeed a triangle where each side has a different length.

Answer

Yes

Exercise #4

Is the triangle in the drawing a right triangle?

Step-by-Step Solution

Due to the presence of the 90 degree angle symbol we can determine that this is indeed a right-angled triangle.

Answer

Yes

Exercise #5

In an isosceles triangle, what are each of the two equal sides called ?

Step-by-Step Solution

In an isosceles triangle, there are three sides: two sides of equal length and one distinct side. Our task is to identify what the equal sides are called.

To address this, let's review the basic properties of an isosceles triangle:

  • An isosceles triangle is defined as a triangle with at least two sides of equal length.
  • The side that is different in length from the other two is usually called the "base" of the triangle.
  • The two equal sides of an isosceles triangle are referred to as the "legs."

Therefore, each of the two equal sides in an isosceles triangle is called a "leg."

In our problem, we confirm that the correct terminology for these two equal sides is indeed "legs," distinguishing them from the "base," which is the unequal side. This aligns with both the typical definitions and properties of an isosceles triangle.

Thus, the equal sides in an isosceles triangle are known as legs.

Answer

Legs

Questions on the subject

What is an equilateral triangle for children?

It is a geometric figure formed by three equal sides.


Why is the triangle equilateral?

Because its three sides have the same length.


What are the angles of an equilateral triangle?

In an equilateral triangle its interior angles are acute and these measure 60° 60° degrees each.


What is an equilateral, isosceles and scalene triangle?

They are geometric figures with three sides, the first one is characterized by having all its sides equal, the second one by having two equal sides and the third one by not having any equal side.


What is another name for the equilateral triangle?

A regular three-sided polygon.


What are the types of triangles?

If the classification is made with respect to their sides we have three types of triangles: equilateral, isosceles and scalene.

If the classification is made with respect to their angles we also find three types of triangles: acute-angled, right-angled and obtuse-angled.


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