Welcome to this video about triangles
a fundamental geometric figure
with multiple applications in mathematics,
engineering, and science.
Today we’ll dive into the theoretical and practical aspects of
triangles, essential to understanding their importance in various disciplines.
Let’s get started!
In this video, we will address the fundamentals of triangles.
We will begin with their definition and nomenclature, then study their basic and advanced geometric properties.
We will look at how to classify triangles by their sides and angles, and focus on key theorems for right triangles.
Finally, we will explore their practical applications in civil engineering.
A triangle is a geometric figure defined by three non-collinear points connected by line segments.
The points are called vertices, the segments are its sides, and the angles are formed at each vertex.
By convention, we use uppercase letters for the vertices,
lowercase letters for the sides opposite them,
and Greek letters for the angles.
This notation allows us to work with precision on the following concepts.
Triangles have key geometric properties:
1.
The sum of the interior angles is always 180 °.
This means that the measures of the three interior angles of any triangle will always add up to this fixed amount.
2.
The triangle inequality states that any side of a triangle is less than the sum of the other two sides.
3.
Each exterior angle of a triangle equals the sum of the two non-adjacent interior angles.
These properties form the foundation for understanding more complex structures.
The area of a triangle is calculated by multiplying the length of its
base by the height relative to that base and dividing the result by two.
If the lengths of all three sides are known, we can use Heron’s formula
First, calculate the semi-perimeter, which is half the sum of the three sides.
Then, the area is found as the square root of the product of the semi-perimeter and its differences with each side.
When two sides and the angle between them are known,
the area is calculated as half the product of those sides and the sine of the angle.
The area can also be calculated using the radii of the circles associated with the triangle:
Using the radius of the inscribed circle, the area is the product of the semi-perimeter and this radius.
On the other hand, using the radius of the circumscribed circle,
the area is the product of the three sides divided by four times the circumscribed radius.
In a triangle, we find remarkable points that define essential geometric properties:
Centroid (G):
This is the point where the medians of the triangle intersect,
meaning the lines that connect each vertex with the midpoint of the opposite side.
This point divides each median in a 2:1 ratio.
Orthocenter (O r):
This is the point where the altitudes of the triangle intersect.
The altitudes are the perpendicular lines drawn from each vertex to the opposite side or its extension.
Circumcenter (C c):
This is the point equidistant from the three vertices of the triangle, and it is the center of the circumscribed circle.
This circle passes through all three vertices of the triangle.
Incenter (I c):
This is the point where the internal angle bisectors of the triangle intersect.
This point is the center of the inscribed circle, which touches all three sides of the triangle.
Euler line:
In any triangle, the centroid, orthocenter, and circumcenter are aligned along a straight line called the Euler line.
Furthermore, the centroid divides the line into specific proportions.
These geometric properties are not only fascinating theoretically but
also have practical applications in fields such as surveying and structural design.
Triangles can be classified in two main ways: by their sides and by their angles.
1. Classification by sides:
Equilateral Triangle:
All three sides are equal, and all angles measure 60°.
It is the most symmetrical triangle and is often found in structural and decorative designs.
Isosceles Triangle:
Two sides are equal, and the third is different.
The angles opposite the equal sides are also equal.
This type is common in structures where visual and physical balance is required.
Scalene Triangle: All sides have different lengths, and all angles are different.
It represents asymmetrical but functional situations in designs and geometric calculations.
2. Classification by angles:
Acute-angled Triangle:
All angles are less than 90 °.
These triangles often appear in compact and enclosed structures.
Right-angled Triangle:
Has one angle of 90 °.
It is the most studied triangle due to its connection to the
Pythagorean Theorem and its essential role in design and construction.
Obtuse Triangle:
Has one angle greater than 90 °.
These triangles are less common in practical applications but hold theoretical importance.
Understanding these classifications not only helps us identify
specific geometric properties but also allows us to apply the right
triangle to problems in design, construction, and structural analysis.
The Pythagorean Theorem is a fundamental principle for right triangles.
It states that the square of the longest side, called the
hypotenuse, is equal to the sum of the squares of the other two sides, known as the legs.
To visualize this, imagine constructing squares on each side of the triangle.
Decomposing the areas of the squares on the legs, we see that they perfectly fit within the square on the hypotenuse.
This method of graphical decomposition provides a powerful visual proof.
Theorem of the Height
Euclid, in his Elements, described a key property of right triangles known as the Height Theorem.
It states that the height relative to the hypotenuse in a right
triangle is the geometric mean of the two segments into which the hypotenuse is divided.
Theorem of the Leg
Another property attributed to Euclid is the Leg Theorem.
It states that each leg is the geometric mean of the hypotenuse and its projection onto the hypotenuse.
Hypotenuse and height theorem
The third important property of right-angled triangles is the Hypotenuse and height Theorem.
It states that:
The product of the hypotenuse by the height is equal to the product of the legs.
Well, now we will look at the geometric shape of trapezoids,
we will explore how this geometric shape plays an essential role in architecture and engineering.
Throughout the chapter, we will learn from its definition to practical applications,
through its properties and areas.
Index and Objectives.
Before diving into the content, let’s review the index and learning objectives.
This video is divided into six sections:
This video is divided into six sections: introduction,
definition and nomenclature
classification based on angles,
properties,
area calculations, and conclusions.
Our main goal is to understand the key characteristics of trapezoids and their applications in different contexts.
Introduction.
Let’s begin by understanding the importance of trapezoids.
This geometric figure, with its flexibility and functionality, has
been used to address structural and aesthetic challenges in various projects.
For instance, the Jewish Museum in Berlin, designed by Daniel Libeskind,
uses trapezoids in its irregular and symbolic floor plan, its structures, and even its windows.
This demonstrates how geometry can be a powerful resource in architectural design.
Definition and Nomenclature:
A trapezoid is a quadrilateral characterized by having at least one pair of parallel sides, known as bases.
Additionally, it features:
• Oblique sides: the non-parallel sides.
• Height: the perpendicular distance between the bases.
• Diagonals: segments connecting opposite vertices.
• Interior angles: which always add up to 360 degrees.
A special feature of the trapezoid is its median parallel, a segment that passes through the midpoints of the oblique sides.
The segment formed by this median parallel is called the median base, and it has the following properties:
1. It is parallel to the bases of the trapezoid.
2. Its length is equal to the average of the lengths of the bases.
Mathematically:
(Median base) is equal to (Base one plus Base two) divided by two.
Classification of Trapezoids
There are three types of trapezoids based on their interior angles:
1. The right trapezoid, where two of its angles are right angles.
2. The isosceles trapezoid, which has non-parallel sides of equal length and equal adjacent angles to the bases.
3. The scalene trapezoid, where all sides and angles are different.
Basic Properties
Let’s explore the basic properties of trapezoids:
1. Diagonals: The diagonals divide the trapezoid into triangles.
When both diagonals are considered, the trapezoid is divided into
four unequal triangles, but with pairs of equal areas in specific cases.
2. Heights: The height is drawn perpendicularly between the bases.
In some cases, the trapezoid can be decomposed into triangles and parallelograms using these heights.
3. Parallels:
By drawing parallel lines from the vertices of the bases,
the trapezoid can be divided into a parallelogram and a triangle, which simplifies area calculations.
Metric Properties.
Now, let’s analyze the metric properties of trapezoids:
1. Segment JK of the median base
This segment is formed by the intersections of the diagonals with the median line.
The segment JK equals the difference between the bases divided by two.
2. Segments related by parallelism:
By drawing parallels from specific points on the trapezoid, we can identify proportional segments,
such as those relating the oblique sides to the bases.
These relationships help us understand how dimensions are distributed in the trapezoid.
Area Calculations.
The area of a trapezoid can be calculated in different ways depending on the available data:
1. If the bases and height are known:
'The area is calculated as the product of the median bases and the height.'
2. If the diagonals and the angle between them are known:
'The area is calculated as the product of the diagonals and the sine of the angle they form, divided by two.'
3. Using equivalent shapes (from section 5.1c):
'The trapezoid can be transformed into an equivalent rectangle or square to simplify area calculations.
This process involves drawing a median line and using the midpoints of the bases.'
For example:
• Taking the midpoints of the bases,
we transform the trapezoid into an equivalent rectangle,
whose base equals the midsegment length,
and height equals the height of the trapezoid.
• We can then convert this rectangle into an equivalent square
using the mean proportional between the midsegment and the height.
Thus, the area of the trapezoid is equal to that of the equivalent rectangle and also to that of the square.
Mathematically:
Area is equal to the square root of the product of the median base and the height.
Unified Conclusions on Triangles and Trapezoids.
Triangles and trapezoids are not just geometric shapes but essential tools in architecture and civil engineering.
Triangles stand out for their rigidity and stability, making them
the foundation of structures such as trusses, bridges, and towers, ensuring efficient load distribution.
Meanwhile, trapezoids offer design flexibility, optimizing space
and materials while solving practical problems such as calculating areas and volumes in roofs and facades.
Beyond functionality, these geometric shapes also hold great aesthetic and symbolic value.
Examples like the Jewish Museum in Berlin show how trapezoids can
convey profound concepts, while triangles bring dynamism and visual strength to innovative designs.
In urban settings, both are crucial for designing safe and
efficient structures that improve mobility and optimize traffic flow.
Ultimately, mastering triangles and trapezoids not only helps solve
technical challenges but also fosters creativity and innovation in projects that aim to balance form, function, and sustainability.
This makes them indispensable tools for future professionals in architecture and engineering.
Thank you for joining me on this journey through the world of trapezoids.
I hope this video has been useful and interesting.
Remember: behind the simplest geometric shapes lie profound concepts and practical applications.
See you next time!