Euclidean Geometry
Logic and Problem Solving
Euclidean Logical Thinking for Digital Content Creators 1. Fundamental Knowledge for Logical Thinking
Hi, I am Jong-hwa Park
I have been doing talks
and writing books about
logical thinking, creative problem solving, and innovative approaches in major companies
My lectures mainly focus on
breaking free from stereotypes, adapting to change
and achieving innovation through new ideas
So I teach people
how to logically assess
and creatively resolve problems
I have a Ph.D. in mathematics
My colleagues in industrial education
mostly come from humanities
So my background in math is somewhat unique
That must be why people always ask me
Why do we study math?
Math is one of the most critical subjects in school
Why do we study math?
There could be many different answers
I believe that it teaches us how to think
Math teaches us how to analyze logically
break down problems into smaller pieces
and approach issues with different perspectives
The goal of this course is clear
By Euclidean geometry, one of the first mathematical systems
we will learn how to think and apply the skills
to our work for greater effectiveness and efficiency
So why do we have to talk about Euclidean geometry?
We are trying to emulate humanity's first system of studying math
For example, to study calculus
you have to already know the high-school level knowledge
Or it's hard for you to learn calculus by heart
In order to solve problems and learn the skills
you need access to its details
In that sense, Euclidean geometry
only requires a fourth-grader's knowledge
to be able to experience the problem-solving skill
Even without knowledge, you can experience it with the ideas
Therefore, I hope you get to explore problem-solving
as well as creating ideas
Euclidean Geometry
Math, philosophy, and literature
lay their origin in Thales, around 600 BCE
Of course there was math or philosophy before him
The pyramids in Egypt were built 4,500 years before today
as people estimate
This is already 2,000 years before Thales
The pyramids were already there
Therefore, math, religion, philosophy, and literature
must've been already highly developed
However, thy were only systematically organized
in the forms of studies after Thales
and the following philosophers and mathematicians in ancient Greece
That's why we consider Thales the first mathematician
There are a few more important ancient Greet mathematicians
Let's take a look
Do they look familiar?
I am sure some does, and some doesn't
The knowledge that already existed were only organized
due to logic
This logic is very closely related to Euclidean geometry
Euclid lived around 300 BCE
He wrote The Elements, a book compiling existing mathematical knowledge
Math at the time primarily focused on geometry
That's why this is called The Elements
At the time, geometry was needed
for constructing buildings and measuring land
Making a line, measuring the angle
or measuring the width and the area
taught these people what logic was
Through geometry, humanity began to learn systematic logic
For example, how do we calculate the area of a triangle?
Base multiplied by height divided by two
Or 1/2 multiplied by base by height
We all know about this formula
But why does this give us the area?
When answering the why questions
we need logical processes to break it down step by step
This is the essence of logic
This formula works for its area
due to the followings
For a triangle, as you can see
we can make a rectangle
Draw a perpendicular line
from one vertex to the other side
You see that the triangle divides the rectangle into two equal parts
The area of the rectangle is twice that of the triangle
And the area of the rectangle is the base of the triangle multiplied by its height
Therefore, the area of the triangle is 1/2 multiplied by base by height
This makes us understand that even for this case
the area of the filled in triangle has 1/2 that of the rectangle
Let's talk a bit more about the area of the triangle
Let's consider this
For a triangle with an angle greater than 90 degrees
or an obtuse triangle
its area is also 1/2 multiplied by base by height
This can be proven as follows
It's worth taking a close look at this proof
In conclusion, the area of a triangle
is calculated by multiplying its base by its height and then dividing by two
But we need to consider what this height is
For example, let's take these two triangles
They have the same area
Why is that?
It's because they share the same base and height
It's not immediately obvious that the areas are the same
But we can deduce that they are by logic
This is the power of logic
Please remember this equivalence
As we solve problems
we will often be required to think back to this
We will be solving some problems together
The most important thing is confidence and active engagement in thinking
This must also be the priority for you, the digital content creators
who need to invent new things
I will be talking about technical things, such as logical reasoning
and creative ideation
But there is something invisible that's much more important
It's your attitude toward the problem
I'm bad at this
I don't know this
I hate math
These thoughts will prevent you from solving simple problems
Even with knowledge and practice
the problem will feel difficult
That's why you need confidence and active engagement
You need to believe that you can solve the problem to do so
If you let fear dictate your actions
you will end up losing
This applies to digital content creation as well
Confidence and proactive attitude are critical qualities for digital content creators
In this first session, we will talk about problems that require ideas more than knowledge
You already have the knowledge needed to solve them
All you need is the idea to actively engage with it
Let's continue with this mindset
Confidence and Proactivity
Problem 1. What is the area of the red-shaded region in the figure?
Please try solving it yourself
Even if you struggle, spend at least three minutes on it
If it's been three minutes
and you couldn't solve it, move on to the next question
How do we calculate the area of the shaded area?
To find the area of the shaded area
we need to subtract the area of the circle from that of the square
This circle is inside the square
Its diameter is 10, and its radius 5
So we can calculate it as follows
Area of the square minus area of the circle is 10 squared minus 5 squared times pi
Or, 100 minus 25 time pi
Problem 1 is identical to the one we just solved
Why am I saying this?
Try cutting the figure in half through the middle
Then, you'll see that some shapes
and sizes align perfectly
Move the lower-left part to the upper-right
so that the shaded regions are combined together
you get this
This simplifies the problem
to subtracting the circle's area from the square's
As we talked about, the square with a side length of 10 has an area of 10 multiplied 10, which is 100
The circle with a radius of 5 has an area of 25 times pi
Therefore, the shaded area is 100 minus 25 times pi
To solve problems effectively
it's crucial to proactively seek ideas
Let's look at a few more problems
Problem 2. Find the area of the shaded region
What is the are for this?
Try drawing auxiliary lines
and approaching it differently
It is less important to get the answer right
than to try different approaches
That is how you develop problem-solving skills
Let's try the auxiliary line as shown
Then, you see that the shaded are
is exactly half of the total rectangle
The rectangle's area is 100, so the shaded region's area is 50
Let's take one more problem
Pleas pause and find your own answer
before you continue with the video
Problem 3. Find the area of the shaded region
Let's first start by drawing some lines
Through the center point of the shaded region
draw lines horizontally and vertically
Now we see that the shaded regions and the white region
divide the rectangle into equal halves
Therefore, the total area of the rectangle is 10 times 6, which is 60
So the shaded region's area is 30
For problem-solving ideas, you need to be proactive in your approach
Try modifying the problem, and sometimes try drawing lines in different directions
Exploring the problem in diverse ways while not touching the conditions
is the key to problem-solving
Let's move on
Problem 4. Find the areas of the three squares
This presents there squares
To solve, let's express the given length of 25
as shown in the diagram
The three sides of square B
are 25 plus 3 plus 8
So it's 36
Therefore, the side length of square B is 12
Square A's side length is 12 minus 3
which is 9
For that of square C is 12 minus 8
which is 4
Therefore, the total area of the three squares
9 squared plus 12 squared plus 4 squared
Which are 81, 144, and 16 respectively
The answer is 241
Problem 5. A hexagon is divided into two colored sections
Find the ratio of the areas of the pink and blue sections
So the answer is something by something
Let's first divide the hexagon like so
Now, the hexagon is divided
into six rectangles as shown
The pink has the area of four rectangles
while the blue has that of two rectangles
So the ratio of the pink to blue areas is 2 to 1
Logical and Creative Thinking
What is logical and creative thinking?
Logic starts with what already exists and finds the necessary conclusions
Creativity imagines what does not exist and creates something new
Logical processes are in geometry proofs or in understanding certain conclusions
In solving problems, we try rotating
or flipping shapes
or imaging new possibilities, which require creative approaches
We need both logic and creativity
Creative ideas applied logically will be the optimal case
Logic and creativity, these are the essential keywords even for digital content creators
Let's take a few more examples that require creative ideas and logical problem-solving processes
Problem 6. Find the are of the red-shaded section
This looks like you need to know something
like the Pythagorean theorem
However, once you reinterpret and restructure the problem
you will find that it can be solved easily
Try changing the diagonal direction
given in the problem, as shown
Reconfiguring the problem and altering the diagonal direction
allows us to consider an additional square
between the large and red squares
A diagonal of a rectangle divides it in half
Therefore, the area of the square with a side length of 5 minus that of 4
equals the area of the part excluding the red section in the smaller square
The total area of the largest square is five squared, which is 25, and that of the middle square, four squared or 16
The area of the part excluding the middle square is 25 minus 16
which is 9
Similarly ,the area excluding the red part in the square with an area of 16 is also 9
Therefore, the area of the red part is 16 minus 9, which is 7
Problem 7. Identical rectangles are arranged as shown in the figure
Find the area of one rectangle
Try rearranging the given rectangles
as shown here
When they are rearranged
the setup of Problem 7 reminds us of Problem 6
The area of the shaded region is twice the difference
between the area of a square with the side length 6 and another of 2
Which means, 6 times 6 minus 2 times 2, or 64
Since the total area of the four rectangles is 64
the area of one rectangle is 16
Instead of jumping right into calculations
try imagining various scenarios, reconfigure them
and gain ideas for problem-solving
This is how you experience the charm of Euclidean geometry
Learning Euclidean geometry through basic concepts and problem-solving
can be a joyful experience of cognitive enjoyment
Our enjoyment can be divided into emotional and cognitive enjoyment
Watching TV shows or a moving film
brings us emotional enjoyment
In contrast, encountering unexpected ideas
or well-structured hidden rules
brings us the enjoyment of thinking, or cognitive enjoyment
This natural fosters logical thinking
and the experience of generating creative ideas
I hope you also enjoy your work as digital content creators
I also hope that you enjoy emotional and cognitive enjoyment
and consider another kind
that you can provide to the consumers of your content
Math was developed in ancient Greece
partly because democracy began there
Democracy involves rational discussion among those with different opinions
through logically convincing each other of their thoughts
Logic is essential to persuade others and assert one's arguments
Logic is the process of understanding the unknown based on the known
This is very similar to the process of problem-solving
You use what you already know to identify
understand ,and respond to the problem
Through Euclidean geometry, I hope you get to enhance both
logical thinking and problem-solving skills
This is especially critical for digital content creators
Without logic, you cannot persuade others
Without ideas, you cannot leave an impact
Logic and ideas are challenges faced by all creators
Here is another problem that requires idea and logic
Give this one a try
Problem 8. Two triangles of the same size are divided into smaller ones of the same size
they are colored blue and orange, respectively
Which colored area is larger?
Please try the problem yourself before you continue with the video
Though this features shapes, it is not a geometry problem
Think about how much the colored parts occupy in each triangle
The blue section divides it into nine pieces
It's 6/9
which is 2/3
The orange one divides it into 16 pieces
So it is 10/16 or 5/8
Now we can compare 2/3 and 5/8 and see which is larger
This is the calculation you can do
And this tells us that the blue area is bigger
Euclidean Geometry Thales, the first mathematician, systematically organized existing knowledge of math and formed the basis for academic study
Euclid, the author of The Elements, compiled and synthesized math of the time People learned logic through methods of construction and measurements
Mathematical formulas require logical process of building step-by-step reasoning
Confidence and Proactive Attitude Thought techniques include logical thinking and creative ideation
In thought techniques, addressing problems often involves focusing on unseen but more significant aspects You must approach problems with confidence and tackle them proactively
To solve problems, you must actively seek ideas and approach them using various methods You must address the problem actively and in various ways, even when the conditions remain unchanged
Logical and Creative Thoughts Logic: Starting from what exists and finding the conclusions you need
Creativity: Imagining what doesn’t exist and creating something new Both logic and creativity are essential in the process of solving problems
Instead of jumping straight into calculations, one should experience reconstructing various scenarios to generate ideas and reach a solution