Triangle

In this article, we will briefly learn everything necessary about triangles and also practice with some exercises!
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Detailed explanation

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Test yourself on triangle!

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 $3$ sides and the sum of all its angles always equals $180$ 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 $90$ 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 $180 º$ .
In the equilateral triangle -> each angle is $60 º$ degrees.
In the isosceles triangle -> the two base angles are equal and the third completes the $180 º$ .
In the right triangle -> only one angle is $90 º$ and the other two complete the $180 º$ .

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

Exercise:
Given the following angles:
angle $A=80 º$
angle $B=50 º$
angle $C=50 º$

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

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Test your knowledge

Question 1

Complete the sentence:

To find the area of a right triangle, one must multiply ________________ by each other and divide by 2.

Question 2

Angle A equals 90°.
Angle B equals 115°.
Angle C equals 35°.

Can these angles form a triangle?

Question 3

Angle A equals 56°.
Angle B equals 89°.
Angle C equals 17°.

Can these angles make a triangle?

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 $3\cdot side$
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?

Question 1

Calculate the area of the following triangle:

Question 2

Calculate the area of the following triangle:

Question 3

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

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:

Video Solution

Step-by-Step Solution

Due to the fact that AB is perpendicular to BC and forms a 90-degree angle,

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

Hence we can calculate the area as follows:

$\frac{AB\times BC}{2}=\frac{8\times6}{2}=\frac{48}{2}=24$

Answer

24 cm²

Exercise #2

Calculate the area of the following triangle:

Video Solution

Step-by-Step Solution

To find the area of the triangle, we will use the formula for the area of a triangle:

$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$

From the problem:

The length of the base $BC$ is given as 7 units.
The height from point $A$ perpendicular to the base $BC$ is given as 4.5 units.

Substitute the given values into the area formula:

$\text{Area} = \frac{1}{2} \times 7 \times 4.5$

Calculate the expression step-by-step:

$\text{Area} = \frac{1}{2} \times 31.5$

$\text{Area} = 15.75$

Therefore, the area of the triangle is $15.75$ square units. This corresponds to the given choice: $15.75$ .

Answer

15.75

Exercise #3

What is the area of the triangle in the drawing?

Video Solution

Step-by-Step Solution

First, we will identify the data points we need to be able to find the area of the triangle.

the formula for the area of the triangle: height*opposite side / 2

Since it is a right triangle, we know that the straight sides are actually also the heights between each other, that is, the side that measures 5 and the side that measures 7.

We multiply the legs and divide by 2

$\frac{5\times7}{2}=\frac{35}{2}=17.5$

Answer

17.5

Exercise #4

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

Video Solution

Step-by-Step Solution

To calculate the area of the triangle, we will follow these steps:

Identify the base, CB, as 6 units.
Identify the height, AC, as 8 units.
Apply the area formula for a triangle.

Now, let's work through these steps:

The triangle is a right triangle with base $CB = 6$ units and height $AC = 8$ units.

The area of a triangle is determined using the formula:

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}

Substituting the known values, we have:

\text{Area} = \frac{1}{2} \times 6 \times 8

Perform the multiplication and division:

\text{Area} = \frac{1}{2} \times 48 = 24

Therefore, the area of the triangle is $24$ square units.

Answer

Exercise #5

Calculate the area of the triangle below, if possible.

Video Solution

Step-by-Step Solution

To solve this problem, we begin by analyzing the given triangle in the diagram:

While the triangle graphic suggests some line segments labeled with the values "7.6" and "4", it does not confirm these as directly usable as pure base or height without additional proven inter-contextual relationships establishing perpendicularity or side/unit equivalences.

Without a clear base and perpendicular height value, we cannot apply the triangle's area formula $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$ effectively, nor do we have all side lengths for Heron's formula.

Therefore, due to insufficient information that specifically identifies necessary dimensions for area calculations such as clear height to a base or all sides' measures, the area of this triangle cannot be calculated.

The correct answer to the problem, based on insufficient explicit calculable details, is: It cannot be calculated.

Answer

It cannot be calculated.

Check your understanding

Question 1

Find the perimeter of the triangle ABC

Question 2

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

Question 3

Calculate the area of the following triangle:

Start practice

Triangle

Triangle

Test yourself on triangle!

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

Triangle Angles

Area of the Triangle

Perimeter of the triangle

Triangle Congruence

Similarity of Triangles

Examples and exercises with solutions of triangles

Exercise #1

Video Solution

Step-by-Step Solution

Answer

Exercise #2

Video Solution

Step-by-Step Solution

Answer

Exercise #3

Video Solution

Step-by-Step Solution

Answer

Exercise #4

Video Solution

Step-by-Step Solution

Answer

Exercise #5

Video Solution

Step-by-Step Solution

Answer

More Questions

Area of a Triangle

Triangle

Perimeter of a Triangle