# Equilateral triangle - Examples, Exercises and Solutions

## 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 $60°$ degrees (remember that the sum of the angles of a triangle is $180°$ degrees and, therefore, these $180°$ degrees are divided equally by the three angles).

## examples with solutions for equilateral triangle

### Exercise #1

What kid of triangle is given in the drawing?

### Step-by-Step Solution

The measure of angle C is 90°, therefore it is a right angle.

If one of the angles of the triangle is right, it is a right triangle.

Right triangle

### Exercise #2

What kind of triangle is given in the drawing?

### Step-by-Step Solution

As all the angles of a triangle are less than 90° and the sum of the angles of a triangle equals 180°:

$70+70+40=180$

The triangle is isosceles.

Isosceles triangle

### Exercise #3

What kid of triangle is the following

### Step-by-Step Solution

Given that in an obtuse triangle it is enough for one of the angles to be greater than 90°, and in the given triangle we have an angle C greater than 90°,

$C=107$

Furthermore, the sum of the angles of the given triangle is 180 degrees so it is indeed a triangle:

$107+34+39=180$

The triangle is obtuse.

Obtuse Triangle

### Exercise #4

What kind of triangle is given in the drawing?

### Step-by-Step Solution

Given that sides AB and AC are both equal to 9, which means that the legs of the triangle are equal and the base BC is equal to 5,

Therefore, the triangle is isosceles.

Isosceles triangle

### Exercise #5

Which kind of triangle is given in the drawing?

### Step-by-Step Solution

As we know that sides AB, BC, and CA are all equal to 6,

All are equal to each other and, therefore, the triangle is equilateral.

Equilateral triangle

### Exercise #6

What kind of triangle is given here?

### Step-by-Step Solution

Since none of the sides have the same length, it is a scalene triangle.

Scalene triangle

### Exercise #7

What kind of triangle is the following

### Step-by-Step Solution

Since in the given triangle all angles are equal, all sides are also equal.

It is known that in an equilateral triangle the measure of the angles will always be equal to 60° since the sum of the angles in a triangle is 180 degrees:

$60+60+60=180$

Therefore, it is an equilateral triangle.

Equilateral triangle

### Exercise #8

Given an equilateral triangle:

What is its perimeter?

### Step-by-Step Solution

Since the triangle is equilateral, that is, all sides are equal to each other.

The perimeter of the triangle is equal to the sum of all sides together, the perimeter of the triangle in the drawing is equal to:

$5+5+5=15$

15

### Exercise #9

Below is an equilateral triangle:

If the perimeter of the triangle is 33 cm, then what is the value of X?

### Step-by-Step Solution

We know that in an equilateral triangle all sides are equal.

Therefore, if we know that one side is equal to X, then all sides are equal to X.

We know that the perimeter of the triangle is 33.

The perimeter of the triangle is equal to the sum of the sides together.

We replace the data:

$x+x+x=33$

$3x=33$

We divide the two sections by 3:

$\frac{3x}{3}=\frac{33}{3}$

$x=11$

11

### Exercise #10

Look at the isosceles triangle below:

What is its perimeter?

### Step-by-Step Solution

Since we are referring to an isosceles triangle, the two legs are equal to each other.

In the drawing, they give us the base which is equal to 4 and one side is equal to 6, therefore the other side is also equal to 6.

The perimeter of the triangle is equal to the sum of the sides and therefore:

$6+6+4=12+4=16$

16

### Exercise #11

Given the isosceles triangle,

What is its perimeter?

### Step-by-Step Solution

Since the triangle is isosceles, that means its two legs are equal to each other.

Therefore, the base is 7 and the other two sides are 12.

The perimeter of a triangle is equal to the sum of all sides together:

$12+12+7=24+7=31$

31

### Exercise #12

Triangle ABC is a right triangle.

The area of the triangle is 6 cm².

Calculate X and the length of the side BC.

### Step-by-Step Solution

We use the formula to calculate the area of the right triangle:

$\frac{AC\cdot BC}{2}=\frac{cateto\times cateto}{2}$

And compare the expression with the area of the triangle $6$

$\frac{4\cdot(X-1)}{2}=6$

Multiplying the equation by the common denominator means that we multiply by $2$

$4(X-1)=12$

We distribute the parentheses before the distributive property

$4X-4=12$ / $+4$

$4X=16$ / $:4$

$X=4$

We replace $X=4$ in the expression $BC$ and

find:

$BC=X-1=4-1=3$

X=4, BC=3

### Exercise #13

Look at the isosceles triangle below:

The perimeter of the triangle is 50.

What is the value of X?

### Step-by-Step Solution

Since we know the triangle is isosceles, the other side will also be equal to X

Now we can replace the data to calculate X.

The perimeter of the triangle is equal to:

$x+x+5.6=50$

$2x=50-5.6$

$2x=44.4$

We divide both sides by 2:

$\frac{2x}{2}=\frac{44.4}{2}$

$x=22.2$

22.2

### Exercise #14

Look at the following triangle:

The perimeter of the triangle is 17.

What is the value of X?

### Step-by-Step Solution

We know that the perimeter of a triangle is equal to the sum of all sides together, so we replace the data:

$3x+2x+3.5x=17$

$8.5x=17$

Divide the two sections by 8.5:

$\frac{8.5x}{8.5}=\frac{17}{8.5}$

$x=2$

2

### Exercise #15

Below is the Isosceles triangle ABC (AC = AB):

In its interior, a line ED is drawn parallel to CB.

Is the triangle AED also an isosceles triangle?

### Step-by-Step Solution

To demonstrate that triangle AED is isosceles, we must prove that its hypotenuses are equal or that the opposite angles to them are equal.

Given that angles ABC and ACB are equal (since they are equal opposite bisectors),

And since ED is parallel to BC, the angles ABC and ACB alternate and are equal to angles ADE and AED (alternate and equal angles between parallel lines)

Opposite angles ADE and AED are respectively sides AD and AE, and therefore are also equal (opposite equal angles, the legs of triangle AED are also equal)