WEBVTT

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Euclidean Geometry

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Logic and Problem Solving

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Euclidean Logical Thinking for Digital Content Creators
1. Fundamental Knowledge for Logical Thinking

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Hi, I am Jong-hwa Park

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I have been doing talks

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and writing books about

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logical thinking, creative problem solving, and innovative approaches in major companies

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My lectures mainly focus on

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breaking free from stereotypes, adapting to change

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and achieving innovation through new ideas

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So I teach people

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how to logically assess

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and creatively resolve problems

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I have a Ph.D. in mathematics

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My colleagues in industrial education

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mostly come from humanities

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So my background in math is somewhat unique

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That must be why people always ask me

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Why do we study math?

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Math is one of the most critical subjects in school

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Why do we study math?

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There could be many different answers

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I believe that it teaches us how to think

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Math teaches us how to analyze logically

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break down problems into smaller pieces

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and approach issues with different perspectives

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The goal of this course is clear

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By Euclidean geometry, one of the first mathematical systems

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we will learn how to think and apply the skills

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to our work for greater effectiveness and efficiency

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So why do we have to talk about Euclidean geometry?

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We are trying to emulate humanity's first system of studying math

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For example, to study calculus

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you have to already know the high-school level knowledge

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Or it's hard for you to learn calculus by heart

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In order to solve problems and learn the skills

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you need access to its details

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In that sense, Euclidean geometry

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only requires a fourth-grader's knowledge

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to be able to experience the problem-solving skill

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Even without knowledge, you can experience it with the ideas

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Therefore, I hope you get to explore problem-solving

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as well as creating ideas

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Euclidean Geometry

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Math, philosophy, and literature

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lay their origin in Thales, around 600 BCE

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Of course there was math or philosophy before him

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The pyramids in Egypt were built 4,500 years before today

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as people estimate

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This is already 2,000 years before Thales

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The pyramids were already there

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Therefore, math, religion, philosophy, and literature

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must've been already highly developed

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However, thy were only systematically organized

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in the forms of studies after Thales

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and the following philosophers and mathematicians in ancient Greece

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That's why we consider Thales the first mathematician

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There are a few more important ancient Greet mathematicians

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Let's take a look

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Do they look familiar?

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I am sure some does, and some doesn't

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The knowledge that already existed were only organized

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due to logic

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This logic is very closely related to Euclidean geometry

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Euclid lived around 300 BCE

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He wrote The Elements, a book compiling existing mathematical knowledge

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Math at the time primarily focused on geometry

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That's why this is called The Elements

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At the time, geometry was needed

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for constructing buildings and measuring land

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Making a line, measuring the angle

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or measuring the width and the area

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taught these people what logic was

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Through geometry, humanity began to learn systematic logic

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For example, how do we calculate the area of a triangle?

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Base multiplied by height divided by two

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Or 1/2 multiplied by base by height

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We all know about this formula

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But why does this give us the area?

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When answering the why questions

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we need logical processes to break it down step by step

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This is the essence of logic

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This formula works for its area

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due to the followings

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For a triangle, as you can see

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we can make a rectangle

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Draw a perpendicular line

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from one vertex to the other side

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You see that the triangle divides the rectangle into two equal parts

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The area of the rectangle is twice that of the triangle

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And the area of the rectangle is the base of the triangle multiplied by its height

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Therefore, the area of the triangle is 1/2 multiplied by base by height

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This makes us understand that even for this case

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the area of the filled in triangle has 1/2 that of the rectangle

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Let's talk a bit more about the area of the triangle

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Let's consider this

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For a triangle with an angle greater than 90 degrees

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or an obtuse triangle

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its area is also 1/2 multiplied by base by height

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This can be proven as follows

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It's worth taking a close look at this proof

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In conclusion, the area of a triangle

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is calculated by multiplying its base by its height and then dividing by two

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But we need to consider what this height is

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For example, let's take these two triangles

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They have the same area

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Why is that?

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It's because they share the same base and height

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It's not immediately obvious that the areas are the same

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But we can deduce that they are by logic

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This is the power of logic

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Please remember this equivalence

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As we solve problems

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we will often be required to think back to this

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We will be solving some problems together

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The most important thing is confidence and active engagement in thinking

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This must also be the priority for you, the digital content creators

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who need to invent new things

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I will be talking about technical things, such as logical reasoning

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and creative ideation

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But there is something invisible that's much more important

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It's your attitude toward the problem

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I'm bad at this

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I don't know this

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I hate math

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These thoughts will prevent you from solving simple problems

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Even with knowledge and practice

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the problem will feel difficult

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That's why you need confidence and active engagement

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You need to believe that you can solve the problem to do so

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If you let fear dictate your actions

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you will end up losing

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This applies to digital content creation as well

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Confidence and proactive attitude are critical qualities for digital content creators

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In this first session, we will talk about problems that require ideas more than knowledge

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You already have the knowledge needed to solve them

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All you need is the idea to actively engage with it

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Let's continue with this mindset

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Confidence and Proactivity

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Problem 1. What is the area of the red-shaded region in the figure?

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Please try solving it yourself

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Even if you struggle, spend at least three minutes on it

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If it's been three minutes

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and you couldn't solve it, move on to the next question

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How do we calculate the area of the shaded area?

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To find the area of the shaded area

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we need to subtract the area of the circle from that of the square

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This circle is inside the square

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Its diameter is 10, and its radius 5

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So we can calculate it as follows

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Area of the square minus area of the circle is 10 squared minus 5 squared times pi

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Or, 100 minus 25 time pi

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Problem 1 is identical to the one we just solved

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Why am I saying this?

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Try cutting the figure in half through the middle

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Then, you'll see that some shapes

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and sizes align perfectly

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Move the lower-left part to the upper-right

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so that the shaded regions are combined together

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you get this

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This simplifies the problem

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to subtracting the circle's area from the square's

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As we talked about, the square with a side length of 10 has an area of 10 multiplied 10, which is 100

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The circle with a radius of 5 has an area of 25 times pi

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Therefore, the shaded area is 100 minus 25 times pi

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To solve problems effectively

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it's crucial to proactively seek ideas

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Let's look at a few more problems

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Problem 2. Find the area of the shaded region

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What is the are for this?

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Try drawing auxiliary lines

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and approaching it differently

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It is less important to get the answer right

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than to try different approaches

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That is how you develop problem-solving skills

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Let's try the auxiliary line as shown

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Then, you see that the shaded are

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is exactly half of the total rectangle

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The rectangle's area is 100, so the shaded region's area is 50

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Let's take one more problem

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Pleas pause and find your own answer

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before you continue with the video

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Problem 3. Find the area of the shaded region

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Let's first start by drawing some lines

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Through the center point of the shaded region

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draw lines horizontally and vertically

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Now we see that the shaded regions and the white region

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divide the rectangle into equal halves

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Therefore, the total area of the rectangle is 10 times 6, which is 60

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So the shaded region's area is 30

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For problem-solving ideas, you need to be proactive in your approach

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Try modifying the problem, and sometimes try drawing lines in different directions

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Exploring the problem in diverse ways while not touching the conditions

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is the key to problem-solving

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Let's move on

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Problem 4. Find the areas of the three squares

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This presents there squares

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To solve, let's express the given length of 25

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as shown in the diagram

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The three sides of square B

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are 25 plus 3 plus 8

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So it's 36

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Therefore, the side length of square B is 12

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Square A's side length is 12 minus 3

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which is 9

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For that of square C is 12 minus 8

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which is 4

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Therefore, the total area of the three squares

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9 squared plus 12 squared plus 4 squared

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Which are 81, 144, and 16 respectively

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The answer is 241

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Problem 5. A hexagon is divided into two colored sections

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Find the ratio of the areas of the pink and blue sections

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So the answer is something by something

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Let's first divide the hexagon like so

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Now, the hexagon is divided

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into six rectangles as shown

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The pink has the area of four rectangles

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while the blue has that of two rectangles

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So the ratio of the pink to blue areas is 2 to 1

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Logical and Creative Thinking

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What is logical and creative thinking?

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Logic starts with what already exists and finds the necessary conclusions

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Creativity imagines what does not exist and creates something new

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Logical processes are in geometry proofs or in understanding certain conclusions

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In solving problems, we try rotating

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or flipping shapes

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or imaging new possibilities, which require creative approaches

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We need both logic and creativity

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Creative ideas applied logically will be the optimal case

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Logic and creativity, these are the essential keywords even for digital content creators

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Let's take a few more examples that require creative ideas and logical problem-solving processes

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Problem 6. Find the are of the red-shaded section

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This looks like you need to know something

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like the Pythagorean theorem

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However, once you reinterpret and restructure the problem

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you will find that it can be solved easily

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Try changing the diagonal direction

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given in the problem, as shown

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Reconfiguring the problem and altering the diagonal direction

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allows us to consider an additional square

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between the large and red squares

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A diagonal of a rectangle divides it in half

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Therefore, the area of the square with a side length of 5 minus that of 4

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equals the area of the part excluding the red section in the smaller square

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The total area of the largest square is five squared, which is 25, and that of the middle square, four squared or 16

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The area of the part excluding the middle square is 25 minus 16

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which is 9

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Similarly ,the area excluding the red part in the square with an area of 16 is also 9

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Therefore, the area of the red part is 16 minus 9, which is 7

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Problem 7. Identical rectangles are arranged as shown in the figure

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Find the area of one rectangle

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Try rearranging the given rectangles

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as shown here

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When they are rearranged

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the setup of Problem 7 reminds us of Problem 6

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The area of the shaded region is twice the difference

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between the area of a square with the side length 6 and another of 2

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Which means, 6 times 6 minus 2 times 2, or 64

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Since the total area of the four rectangles is 64

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the area of one rectangle is 16

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Instead of jumping right into calculations

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try imagining various scenarios, reconfigure them

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and gain ideas for problem-solving

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This is how you experience the charm of Euclidean geometry

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Learning Euclidean geometry through basic concepts and problem-solving

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can be a joyful experience of cognitive enjoyment

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Our enjoyment can be divided into emotional and cognitive enjoyment

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Watching TV shows or a moving film

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brings us emotional enjoyment

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In contrast, encountering unexpected ideas

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or well-structured hidden rules

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brings us the enjoyment of thinking, or cognitive enjoyment

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This natural fosters logical thinking

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and the experience of generating creative ideas

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I hope you also enjoy your work as digital content creators

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I also hope that you enjoy emotional and cognitive enjoyment

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and consider another kind

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that you can provide to the consumers of your content

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Math was developed in ancient Greece

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partly because democracy began there

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Democracy involves rational discussion among those with different opinions

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through logically convincing each other of their thoughts

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Logic is essential to persuade others and assert one's arguments

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Logic is the process of understanding the unknown based on the known

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This is very similar to the process of problem-solving

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You use what you already know to identify

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understand ,and respond to the problem

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Through Euclidean geometry, I hope you get to enhance both

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logical thinking and problem-solving skills

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This is especially critical for digital content creators

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Without logic, you cannot persuade others

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Without ideas, you cannot leave an impact

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Logic and ideas are challenges faced by all creators

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Here is another problem that requires idea and logic

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Give this one a try

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Problem 8. Two triangles of the same size are divided into smaller ones of the same size

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they are colored blue and orange, respectively

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Which colored area is larger?

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Please try the problem yourself before you continue with the video

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Though this features shapes, it is not a geometry problem

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Think about how much the colored parts occupy in each triangle

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The blue section divides it into nine pieces

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It's 6/9

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which is 2/3

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The orange one divides it into 16 pieces

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So it is 10/16 or 5/8

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Now we can compare 2/3 and 5/8 and see which is larger

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This is the calculation you can do

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And this tells us that the blue area is bigger

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Euclidean Geometry
Thales, the first mathematician, systematically organized existing knowledge of math and formed the basis for academic study

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Euclid, the author of The Elements, compiled and synthesized math of the time
People learned logic through methods of construction and measurements

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Mathematical formulas require logical process of building step-by-step reasoning

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Confidence and Proactive Attitude
Thought techniques include logical thinking and creative ideation

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In thought techniques, addressing problems often involves focusing on unseen but more significant aspects
You must approach problems with confidence and tackle them proactively

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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

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Logical and Creative Thoughts
Logic: Starting from what exists and finding the conclusions you need

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Creativity: Imagining what doesn’t exist and creating something new
Both logic and creativity are essential in the process of solving problems

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Instead of jumping straight into calculations, one should experience reconstructing various scenarios to generate ideas and reach a solution