WEBVTT

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Euclidean Geometry

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Logic and Problem-Solving

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Logical Thinking in Euclidean Geometry for Digital Content Creators
5 Finding Ideas with Euclidean Thinking 3. Identifying Known Shapes

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Hello, this is Park Jongha

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Today, let's look at the most important keyword

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in problem-solving: connection

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What is studying?

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It's learning what you don't know

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Learning what you don't know means

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understanding and remembering what you don't know

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by connecting it to what you already know

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If you understand the relationship with something you know well and are familiar with

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you will understand what you don't know and remember it for a long time

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However, if you don't connect what you don't know to what you know

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and just try to accept it

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you won't understand it, and it will be difficult to remember for a long time

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This is logic

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Logic is understanding what you don't know with what you know

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Persuasion is connecting what the other person knows

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with what you are arguing

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Problem-solving is the same

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The only way to solve a problem you don't know is

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to connect what you know with what you don't know

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This is something that is necessary for you, the digital content creators

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and this time, we will look at this content

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Finding Familiar Elements

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I emphasized that problem-solving is finding the answer

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by connecting what you don't know with what you know

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So when you come across a problem that you don't know well and are unfamiliar with

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you should look for something you already know well and are familiar with

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Find something you know and figure out the relationship with what you don't know

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That's because, through that process, you find the answer to the problem you don't know

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Let's take a closer look at the problem

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Question 1. Find the area of ​​the given triangle

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To solve this problem, let's think about the familiar

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30 degrees, 60 degrees, and 90 degrees right triangles

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Since the given angle in the problem is 30 degrees

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let's draw a 30 degrees, 60 degrees, and 90 degrees right triangle as shown below

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30 degrees, 60 degrees, and 90 degrees right triangles

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are made from equilateral triangles

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If we divide an equilateral triangle in half

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we get 30 degrees, 60 degrees, and 90 degrees right triangles

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The thing to note here is the ratio of the lengths

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The length of the hypotenuse of a right triangle is twice the length of the shorter side

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If we apply the ratio of the lengths of 30 degrees, 60 degrees, and 90 degrees right triangles

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to the situation in the problem we looked at earlier

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we can see that the height of the given triangle is 4

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because the hypotenuse is 8

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To summarize, the triangle in the question

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has a base of 6 and a height of 4

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Therefore, the area of ​​the triangle is 6 x 4 x 1/2

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which is 12

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The ratios of the lengths of the 30 degrees, 60 degrees, and 90 degrees right triangles made by cutting an equilateral triangle in half

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are crucial when solving Euclidean

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If we apply the Pythagorean theorem, we can see that the ratio of the lengths of these triangles

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is 1 : root of 3 : 2

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but since we are not dealing with the Pythagorean theorem

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we'll not use the 1 : root of 3 : 2 ratio

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and will only use the fact that

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the ratio of the lengths of the hypotenuse and the shorter side is 2 to 1, as we have just seen

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Another right triangle that is frequently used

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is the 45 degrees, 45 degrees, and 90 degrees right triangles

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which are made by cutting

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a square in half diagonally

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The ratio of the lengths of these right triangles

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is often used to mean that the lengths of the two sides, except the hypotenuse, are equal

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It would be good to look at how these right triangles are used

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while solving problems

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Question 2. Find the value of x in the given figure below

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This problem is complicated

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but let's think about the original pieces that were cut off

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Imagine the pieces that might have been cut off

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So, let's try to attach the pieces that might have been cut off

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The angles of A and E are 120 degrees

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so 180 degrees minus 120 degrees is 60 degrees

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Thinking about this, we can think about

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attaching the following equilateral triangles in addition to AE

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After attaching this equilateral triangle to AE

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we can attach right triangles with 30 degrees, 60 degrees, and 90 degrees to the remaining BC

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to make the entire thing one large 30 degrees, 60 degrees, and 90 degrees right triangle

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Since the entire large right triangle GDF is a 30 degrees, 60 degrees, and 90 degrees right triangle

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the ratio of the hypotenuse length to the shorter side is 2:1

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DF is 5 + 7

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which is 12

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so GF is 24

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Since GB is 6 and AF is 7

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GF is GB plus x plus AF

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which is 6 + x + 7

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which is 24

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Therefore, x is 11

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Let's look at another question

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Question 3. Given the figure below, find the length of AB

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If you imagine this problem is similar to Problem 2 with one part cut off

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and think about the original shape before it was cut off

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you can think of an equilateral triangle OBC, as you can see now

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The cut-off part is

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a right triangle OAD with angles of 30, 60, and 90 degrees

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Since OBC is an equilateral triangle

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OD is 2

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and the right triangle OAD is a right triangle with angles of 30, 60, and 90 degrees

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OA is 4

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Therefore, the answer is x + 4 is 12

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and x is 8

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As we have seen above

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right triangles, created by dividing equilateral triangles and squares in half

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are useful in Euclidean geometry

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To summarize

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we can think of two types of right triangles

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a 30 degrees, 60 degrees, 90 degrees right triangle

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and a 45 degrees, 45 degrees, 90 degrees right triangle

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and the ratio of the lengths of the two right triangles is as follows

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Let's solve the problem again

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using these two right triangles

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Question 4. Find the area of ​​an isosceles triangle

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where AB is 8, AC is 8, and angle B is 75 degrees

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If we look into the information about the given shape a little more

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since angle B is 75 degrees, angle C is also 75 degrees

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and angle A is 180 degrees - 75 degrees x 2

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- 150 degrees, which is 30 degrees

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Looking at the information written like this

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we need to think about angle B as 75 degrees

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as 60 degrees plus 15 degrees

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Let's draw line segment BD

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from point B to AC to find point D

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where the angle ABD is 60 degrees

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If we draw lines like this, we can see that triangle ABD

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is a familiar 30 degrees, 60 degrees, and 90 degrees

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right triangle

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The ratio of the lengths of 30 degrees, 60 degrees, and 90 degrees right triangles

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is AB : BD, which is 2:1

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So BD is 4

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and BD is the height of hypotenuse AC in triangle ABC

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In conclusion, the area of ​​triangle ABC can be calculated as follows

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1/2 x 4 x 8

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which is 16

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Let me emphasize again that the only way to figure out what you don't know

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is to approach what you don't know through what you know

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In Euclidean geometry, you don't need to know much

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If you can utilize just the special right triangles 30 degrees, 60 degrees, and 90 degrees

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and right triangles 45 degrees, 45 degrees, and 90 degrees

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you can solve many problems

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Finding the special right triangles we need is the key to solving the problem

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Here's another question

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Question 5. What is the area of ​​a right triangle with a hypotenuse of 2 and one angle of 15 degrees?

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The problem is about a right triangle with an angle of 15 degrees

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As I mentioned, we know right triangles with angles of 30 degrees, 60 degrees, and 90 degrees

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but we are unfamiliar with 15 degrees

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Let's think about how we can utilize the right triangles

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with angles of 30 degrees, 60 degrees, and 90 degrees that we know

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Let's think about a triangle whose vertex is an angle of 30 degrees

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by copying the given triangle and pasting it to the side, as shown below

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If we think about a triangle with one angle of 30 degrees like this

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we can find a special right triangle with angles of

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30 degrees, 60 degrees, and 90 degrees

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The ratio of the hypotenuse length and the shorter side of this right triangle is 2:1

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So the area of ​​triangle ACC' is

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1/2 x 2x 1

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which is 1

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The area of ​​triangle ABC that we are looking for will be half of this

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1/2 is the answer

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Calculating and Imagining

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When we look at a math problem

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we first think about

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how to calculate the x and y variables

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However, before doing such calculations

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we need to think and imagine something

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When we think of math, we often think of complex calculations

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but that is not true math

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The process of thinking about various things and imagining new things is

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the process of first going through that process

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and then processing it, which is what calculations are needed for

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I think this applies similarly to you, digital content creators

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Rather than just sitting in front of a computer and starting work

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it would be more effective first to imagine

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what kind of work you will do and how you will do it

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We are currently learning Euclidean geometry from ancient Greece

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but the ancient Greeks did not have calculation skills

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The decimal system that we use today is a very advanced form of positional description

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We can easily do various calculations, not just the four basic operations

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but the ancient Greeks found even multiplication and division very difficult

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So, I think they spent more time thinking logically while imagining things rather than calculating

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And such imagination and logic

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became the cornerstones of today’s civilization

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For example, let's think about this problem

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What is the area of ​​the shaded area of ​​the square

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with a diagonal length of 10 in the following figure?

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If you ask middle and high school students about this problem, most of them will do the calculation

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How did the ancient Greeks, who could not calculate, solve this problem?

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So, I think they would have solved this problem

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logically using their imagination

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Let's think about copying the shape given in the question

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and pasting it like this

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If you paste it like this, you can see that the area we are looking for

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is one-fourth of a square with a side length of 10

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The area of ​​a square with a side length of 10 is 100

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The  ​​shaded area is 25

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You should do the calculation when necessary

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However, it is a good way to solve the problem

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to imagine before calculating

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and approaching it in various ways

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It is not a good habit to just calculate

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Here's a question

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Question 6. Find the area of ​​the given shape

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The given shape can be considered a square

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and a 30 degrees, 60 degrees, and 90 degrees right triangle

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Let's divide the 30 degrees, 60 degrees, and 90 degrees right triangles

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in the problem into four smaller

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30 degrees, 60 degrees, and 90 degrees right triangles

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If the two sides of the 30 degrees, 60 degrees, and 90 degrees right triangles are AB

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the hypotenuse will be 2a

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The hypotenuse of the smaller 30 degrees, 60 degrees, and 90 degrees right triangle created in this way

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is the same length as one side of the left square

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Therefore, if you put them together

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the entire thing becomes one square

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The condition of the problem is that 2a + 2b is 6

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The side length of a new square made by cutting and pasting

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four small rectangles is a + b

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which is 3

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Therefore, the area of ​​the newly created square is

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3 squared

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9

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The area of ​​the shape we are looking for is 9

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Solving the same problem differently

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is a great way to gain different perspectives

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and think of new things

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Let's solve this problem in another way

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First, let's look at the following picture

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If you look at a right isosceles triangle

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and an isosceles triangle with an angle of 30 degrees

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we can see that when the two shapes have the same base

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their heights are also the same

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If the bases of the two triangles are 2a

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the heights of the two triangles are both a, which is the same

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Since the bases and heights are the same, the areas of the two triangles will also be the same

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Let's apply this fact to this problem

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If you draw a line parallel to the line connecting AD

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once through point A

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and if you call the point where that line intersects the extension of CD, A'

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triangle BA'D becomes an isosceles triangle with angle A'BD of 30 degrees

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Since the areas of the right isosceles triangle ABD

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and the isosceles triangle BA'D are equal

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Question 6 can now be solved by finding the area

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of ​​the larger isosceles triangle CBA'

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In summary, this problem has changed to finding the area of ​​an isosceles triangle

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with a side length of 6 and an angle of 30 degrees

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We have already solved finding the area of ​​an isosceles triangle with an angle of 30 degrees

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in Question 4

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If we solve it using the same method, we can find

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30 degrees, 60 degrees, and 90 degrees right triangles

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We can see that it is an isosceles triangle with a base of 6 and a height of 3

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Therefore, the area is 6 x 3 x 1/2

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is 9

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I will introduce one more problem that requires an idea and amazing insight

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It is a problem that requires you to find a familiar shape

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When you see the solution

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you will exclaim

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Wow, there's something like this too

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Give it a try

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Question 7. Find the area of ​​the shaded triangle in the following square

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This problem makes me think that there is too little information given

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It is a bit absurd

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It is necessary to pay attention to the fact that a square and a right triangle are given

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You can copy the right triangle of the square

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and paste four of them inside

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Here, we need to look at the two right triangles

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related to the shaded part

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The shaded triangle ADE can be understood as a triangle with a base AB of length 6

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and a height of 6

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Does it catch your eye?

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I think it would be good to understand by looking at this picture

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Therefore, the area is 6 x 6 x 1/2

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the answer is 18

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If you take a math test, you need to answer quickly, calculate accurately

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and write the answer within the given time

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However, it is not important to quickly come up with one answer when studying math

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If you only come up with one answer quickly, there are many cases where you do not study much

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If you think about one problem in various ways and solve the problem while thinking deeply

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you will study well

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In that sense, rather than simply calculating

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thinking about various things and logically examining them

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will be a real study

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I think this attitude is necessary for those who create digital content

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Rather than just making something that is set

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the basic attitude of a creator is to think about something else

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and approach it more diversely

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If you think of it in terms of art and business

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a digital content creator does art plus business

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If producing results is business

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then thinking about various things in various ways is art

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I hope you become a competent digital content creator who catches two rabbits

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Finding the Familiar
When faced with an unfamiliar problem, you should find something you already know well and are familiar with it and figure out the relationship between it and the unknown

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To figure out the unknown, you should approach the unknown through what you know
Most problems can be solved through right triangles of 30 degrees, 60 degrees, and 90 degrees and right triangles of 45 degrees, 45 degrees, and 90 degrees

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Calculating and imagining
Before calculating to solve a problem, a process of thinking and imagining is necessary

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Mathematics is not about doing complex calculations but rather performing calculations in the process of processing various thoughts and new imaginations

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Solving the same problem in various ways allows you to have various perspectives and think of new things
You must have an attitude of thinking about various things and logically examining them