Triangle

In this article, we will briefly learn everything necessary about triangles and also practice with some exercises!
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Test yourself on triangle!

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Angle A is equal to 30°.
Angle B is equal to 60°.
Angle C is equal to 90°.

Can these angles form a triangle?

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The Parts of a Triangle - What Is It Made Of?

  • Line: It is the union of a sequence of points, which are located in a linear way, that is, there are no curves between them.
  • Segment: It is a portion of a line, which is joined between two points.
  • Height: The height of a triangle is the measure or length from a vertex to the highest point of the triangle, it is usually denoted with the letter h.
  • Median: the median is the segment that extends from a given vertex to the midpoint of the opposite side to that vertex.
  • Bisector: the bisector is a ray that extends from a given vertex, dividing it into two equal angles.
  • Perpendicular bisector: the perpendicular bisector is the line that exactly halves its sides and can be drawn perpendicular to those sides.
  • Middle segment: In the case of triangles, the middle segment is that line that we can draw by locating the midpoint of two sides, this line will measure half of the third side.
  • Opposite side: an opposite side is one that is located in front of a given vertex.


The triangle is a figure composed of 33 sides and the sum of all its angles always equals 180180 degrees.
There are several types of triangles:


Equilateral triangle - All sides (or edges) are equal, all angles are equal, and all heights are also the median and the bisector.
Isosceles triangle - It has two equal sides, two equal base angles, and the median is also the height and the bisector.
Right triangle - It has an angle of 9090 degrees formed by two legs. The side opposite the right angle is called the hypotenuse.
Scalene triangle - All sides of the triangle are different.

Click here for a more in-depth explanation about the types of triangles.


Triangle Angles

In any triangle, regardless of the type of triangle it is, the sum of all its angles equals 180º180 º .
In the equilateral triangle -> each angle is 60º60 º degrees.
In the isosceles triangle -> the two base angles are equal and the third completes the 180º180 º .
In the right triangle -> only one angle is 90º90 º and the other two complete the 180º180 º .

Another note:
In the special triangle of 90 º , 45 º , 45 º -> only one angle is 90º90 º and the other two are 45º45 º each, this creates a triangle that is both isosceles and right at the same time.

Exercise:
Given the following angles:
angle A=80ºA=80 º
angle B=50ºB=50 º
angle C=50ºC=50 º

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Area of the Triangle

Here we will present the general formula for calculating the area of triangles:

A3 - the general formula for calculating the area of triangles

This formula is used to calculate the area of isosceles, equilateral, and scalene triangles.
Right triangle

length of the first leg × \times length of the second leg
\frac{length~of~the~first~leg~\times ~length~of~the~second~leg

Click here for a more in-depth explanation about the area of the triangle.


Perimeter of the triangle

The perimeter of the triangle is equal to the sum of the lengths of all sides.

In an equilateral triangle – all sides are equal, therefore, the perimeter of the triangle will be 3side3\cdot side
In an isosceles triangle - there are two equal sides and it is convenient to remember this when we want to deduce the perimeter

Click here for a more in-depth explanation about the perimeter of the triangle.


Do you know what the answer is?

Triangle Congruence

Triangles are considered congruent if all their angles and all their sides are equal respectively.
To prove that two triangles are congruent you must demonstrate one of the following congruence theorems:

ASA – angle, side, angle
If both triangles have 2 equal angles and the length of the side between them is also equal, the triangles are congruent.

SAS – side, angle, side
If both triangles have 2 equal sides and the adjacent angle is also equal, the triangles are congruent.

SSS - Side, side, side
If the lengths of all 3 sides are equal respectively in both triangles, the triangles are congruent.

SSA - Side, side, angle
If the 2 sides are equal in both triangles and so is the angle opposite the larger side, the triangles are congruent.

Click here for a more in-depth explanation on triangle congruence.


Similarity of Triangles

Similar triangles do not need to have identical areas as is the case with congruent triangles, it is enough that they have the same proportions.
To prove that two triangles are similar you must demonstrate one of the following similarity theorems:

AA – Angle, Angle
If two angles of one triangle are equal to two angles of the other, the triangles are similar.

SSS - Side, Side, Side
If in one triangle the three sides are proportional to the three sides of the other, the triangles are similar.

Click here for a more in-depth explanation on the similarity of triangles.


Examples and exercises with solutions of triangles

Exercise #1

Calculate the area of the right triangle below:

101010666888AAACCCBBB

Video Solution

Step-by-Step Solution

As we see that AB is perpendicular to BC and forms a 90-degree angle

It can be argued that AB is the height of the triangle.

Then we can calculate the area as follows:

AB×BC2=8×62=482=24 \frac{AB\times BC}{2}=\frac{8\times6}{2}=\frac{48}{2}=24

Answer

24 cm²

Exercise #2

Calculate the area of the triangle ABC using the data in the figure.

121212888999AAABBBCCCDDD

Video Solution

Step-by-Step Solution

First, let's remember the formula for the area of a triangle:

(the side * the height that descends to the side) /2

 

In the question, we have three pieces of data, but one of them is redundant!

We only have one height, the line that forms a 90-degree angle - AD,

The side to which the height descends is CB,

Therefore, we can use them in our calculation:

CB×AD2 \frac{CB\times AD}{2}

8×92=722=36 \frac{8\times9}{2}=\frac{72}{2}=36

Answer

36 cm²

Exercise #3

Calculate the area of the following triangle:

444555AAABBBCCCEEE

Video Solution

Step-by-Step Solution

The formula for calculating the area of a triangle is:

(the side * the height from the side down to the base) /2

That is:

BC×AE2 \frac{BC\times AE}{2}

Now we replace the existing data:

4×52=202=10 \frac{4\times5}{2}=\frac{20}{2}=10

Answer

10

Exercise #4

Calculate the area of the triangle below, if possible.

8.58.58.5777

Video Solution

Step-by-Step Solution

The formula to calculate the area of a triangle is:

(side * height corresponding to the side) / 2

Note that in the triangle provided to us, we have the length of the side but not the height.

That is, we do not have enough data to perform the calculation.

Answer

Cannot be calculated

Exercise #5

Given the triangle:

777111111131313

What is its perimeter?

Video Solution

Step-by-Step Solution

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

11+7+13=11+20=31 11+7+13=11+20=31

Answer

31

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