The sides or edges of a triangle

The sides of a triangle

Every triangle has three sides. That also works the other way around - if we see a shape with tree sides, it's a triangle.

types of triangles based on the sides:

The sides allow us to classify the different types of triangles according to their size:

Equilateral: All sides are equal, leading to equal angles.
Isosceles: Two sides are equal, with base angles also equal.
Scalene: All sides are different lengths, with all angles unique.

Perimeter of a Triangle

Like every polygon, the sides of a triangle form its perimeter. To find the perimeter of a triangle, simply add the lengths of all three sides.

Relation between the sides and the angles in a triangle

In a triangle, there’s a direct relationship between the length of a side and the size of the angle across from it:
The Longer Side will always be in the opposite side of the larger Angle, and the shorter side will always be in the opposite side of the smaller Angle.

Can every three lines form a triangle?

In any triangle, the sum of the two shorter sides must always be greater than the length of the third side. This rule, known as the Triangle Inequality Theorem, ensures that the sides can actually form a closed triangle. For example, if the two shorter sides are not greater than the third, the sides would lie flat rather than forming a triangle. This principle is crucial in determining whether a set of side lengths can create a valid triangle.

Detailed explanation

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Test yourself on parts of a triangle!

True or false:

DE not a side in any of the triangles.

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Perimeter of the triangle

Recall that the perimeter of a plane figure is its edge, so in a triangle the perimeter is the sum of its three sides (edges).

A condition satisfied by the measures of the sides (or edges) of a triangle.

In any triangle the sum of the length of any two of its sides must be greater than the length of the third side.

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Test your knowledge

Question 1

Is DE side in one of the triangles?

Question 2

True or false:

AB is a side of the triangle ABC.

Question 3

True or false:

AD is a side of triangle ABC.

Examples of the subject

Example 1

Given a triangle with sides $4~cm$ , $3~cm$ and $5~cm$ . Calculate the perimeter.

Solution

We know that the perimeter of a triangle is the sum of its three sides, therefore,

$P=4\operatorname{cm}+3\operatorname{cm}+5\operatorname{cm}$

Answer:

$P=12\operatorname{cm}$

Example 2

Tell if it is possible to construct a triangle in which its sides measure $3~cm$ , $4~cm$ and $8~cm$ .

Solution

Recall that in order to construct a triangle, the sum of any two sides must be greater than the third side.

If we add the sides with measures $3~cm$ and $4~cm$ , we get as a result $7~cm$ , which is less than the third side.

Answer:

Therefore, it is not possible to construct a triangle with the given measures.

Do you know what the answer is?

Question 1

True or false:

BC is a side of triangle ABC.

Question 2

ABC is an isosceles triangle.

AD is the median.

What is the size of angle $∢\text{ADC}$ ?

Question 3

AD is the median in triangle ABC.

BD = 4

Find the length of DC.

Example 3

If an equilateral triangle has perimeter $P=21\operatorname{cm}$ . How long is each side?

Solution

Since the triangle is equilateral we know that its sides are equal, so we just divide the perimeter by three to get the measure of each side. lado.

$P=3x$

$21\operatorname{cm}=3x$

$x=21\operatorname{cm}:3$

Answer:

$x=7\operatorname{cm}$

Example 4

Tell if it is possible to construct a triangle in which its sides measure $5~cm$ , $7~cm$ and $10~cm$ .

Solution

We add the lengths of any two sides (edges) and compare with the length of the remaining side.

$5~cm + 7~cm= 12~cm$ which is greater than the remaining side that measures $10~cm$
$7~cm + 10~cm = 17~cm$ The length of the remaining side, which is greater than the remaining side measuring $5~cm$ , is greater than the remaining side measuring $5~cm$ .
$10~cm + 5~cm = 15~cm$ , which is greater than the remaining side measuring $7~cm$ .

Answer:

So if it is possible to construct a triangle of measures $5~cm$ , $7~cm$ and $10~cm$ on each side.

Check your understanding

Question 1

Given the following triangle:

Write down the height of the triangle ABC.

Question 2

Given the following triangle:

Write down the height of the triangle ABC.

Question 3

Given the following triangle:

Write down the height of the triangle ABC.

Questions on the subject

How many sides does a triangle have?

A triangle has three sides.

How many edges does a triangle have?

A triangle has three edges.

What are the edges of a triangle?

The edges of a triangle, commonly called the sides of a triangle, are the straight lines that bound the faces of the triangle.

What are the edges of a figure?

In a plane figure, the edges or sides are the line segments that join two vertices, and form the outline or perimeter of the figure.

Exercises on the sides or edges of a triangle

Exercise 1

Query

$DE$ Does that side not exist as part of any of the triangles?

Consignment DE This side does not exist as part of any of the triangles.

Solution

A side in a triangle is a line that passes between one of the 3 points that are the angles of the triangle.

In this case the line $DE$ does not pass between the extreme angles of any of the triangles but goes out through a point $D$ which is in fact an angle in a triangle $\triangle DBC$ but $DE$ ends at the point $E$ which is not an angle in any of the triangles in the figure.

Answer

True

Do you think you will be able to solve it?

Question 1

Which of the following is the height in triangle ABC?

Question 2

Given the following triangle:

Write down the height of the triangle ABC.

Question 3

Given the following triangle:

Write down the height of the triangle ABC.

Exercise 2

Question:

Do the triangles in the drawing overlap?

Solution

We can observe that according to the theorem of superposition: side, side, angle.

We can observe that there are 2 sides equal in length and an angle equal in size.

Answer

Yes

Exercise 3

Question

What type of triangle is drawn here?

Solution

It can be seen that in this triangle each of its three angles is of different size so it can be said that it is a scalene triangle.

Answer

Scalene triangle

Test your knowledge

Question 1

Determine the type of angle given.

Question 2

Determine the type of angle given.

Question 3

True or false:

DE not a side in any of the triangles.

Exercise 4

Consigna

Given the following triangle:

The perimeter of the triangle is $17$

How much is $X$ ?

Solution

To solve the task we replace all the data we have in the equation to calculate the perimeter of the triangle:

$2X+3X+3.5X=17$

Let's remember... The perimeter of the triangle is equal to the sum of its 3 sides.

If we calculate the equation we find that:

$8.5X=17$

We divide the equation by $8.5$ to find the value of. $X$

$\frac{8.5X}{8.5}=X=\frac{17}{8.5}=2$

Answer

$2$

Exercise 5

Request

Given the equilateral triangle

The perimeter of the triangle is $33\operatorname{cm}$ , what is the value of $X$ ?

Solution

One of the characteristics of an equilateral triangle is obviously that each of its sides are equal, i.e. if one side is worth $11$ all its sides will be equal to $11$

Answer

$11$

Do you know what the answer is?

Question 1

Is DE side in one of the triangles?

Question 2

True or false:

AB is a side of the triangle ABC.

Question 3

True or false:

AD is a side of triangle ABC.

Examples with solutions for The sides or edges of a triangle

Exercise #1

ABC is an isosceles triangle.

AD is the median.

What is the size of angle $∢\text{ADC}$ ?

Video Solution

Step-by-Step Solution

In an isosceles triangle, the median to the base is also the height to the base.

That is, side AD forms a 90° angle with side BC.

That is, two right triangles are created.

Therefore, angle ADC is equal to 90 degrees.

Answer

Exercise #2

Given the following triangle:

Write down the height of the triangle ABC.

Video Solution

Step-by-Step Solution

An altitude in a triangle is the segment that connects the vertex and the opposite side, in such a way that the segment forms a 90-degree angle with the side.

If we look at the image it is clear that the above theorem is true for the line AE. AE not only connects the A vertex with the opposite side. It also crosses BC forming a 90-degree angle. Undoubtedly making AE the altitude.

Answer

Exercise #3

Which of the following is the height in triangle ABC?

Video Solution

Step-by-Step Solution

Let's remember the definition of height of a triangle:

A height is a straight line that descends from the vertex of a triangle and forms a 90-degree angle with the opposite side.

The sides that form a 90-degree angle are sides AB and BC. Therefore, the height is AB.

Answer

Exercise #4

Can a triangle have two right angles?

Video Solution

Step-by-Step Solution

The sum of angles in a triangle is 180 degrees. Since two angles of 90 degrees equal 180, a triangle can never have two right angles.

Answer

Exercise #5

Look at the two triangles below. Is EC a side of one of the triangles?

Video Solution

Step-by-Step Solution

Every triangle has 3 sides, let's go over the triangle on the left side:

Its sides are: AB, BC, CA

This means that in this triangle, side EC does not exist.

Let's go over the triangle on the right side:

Its sides are: ED, EF, FD

This means that in this triangle, side EC does not exist.

Therefore, EC is not a side in either of the triangles.

Answer

No.

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