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
6 Breaking the Mold of the Right Answer - Problem Solving Techniques 1

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Hello, this is Park Jongha

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In this lesson, I will introduce problem-solving techniques

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such as collecting information and imagining in various ways

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In this situation, it is difficult and often useless

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to talk about

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applying these problem-solving techniques in a short story

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The problems we face are

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unlike the math problems we are learning now, so they are not immediately apparent

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However, I think it is very important to think about the need for these things in everyday life

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In that sense, it would be good to remember that collecting sufficient information in this lesson is necessary

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You should think about various cases

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in various ways

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In particular, in the digital content market

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where new technologies and services are created daily

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you should always be careful when collecting new information

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Transforming Given Information into Needed Information

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When solving a problem, collecting all the necessary information is important

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The more information you have

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the easier and faster it is to approach problem-solving

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 If you lack information, you cannot solve the problem

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no matter how many ideas you try

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Therefore, it is effective to approach the problem by indicating information

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that will likely be used to solve the problem

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For example, let's look at this question

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In the following figure, what is x's angle

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when RT and TU are equal?

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Under the condition of the problem that RT is TU

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triangle RTU is an isosceles triangle

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Since 180 degrees - 106 degrees is 74 degrees

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the sizes of the remaining two angles of an isosceles triangle are 37 degrees each

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Let's write this down in the problem

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If you write down the angles in the problem like this

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By the exterior angle theorem

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angle x in triangle SVU is 32 degrees plus 37 degrees

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equals 69 degrees

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Math problems are not solved by eye but by hand

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The first thing you should do by moving your hand

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is to indicate the necessary information

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As you display the information necessary for problem-solving in this way

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you will often have the experience of seeing the information you have displayed with your eyes

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moving your head

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and coming up with ideas

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necessary for problem-solving

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I recommend that those of you who deal with digital content

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also hold a ballpoint pen in your hand on a practice sheet

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like an analog and express your thoughts in various ways

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Writing something on paper with your hand often gives you ideas

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so I recommend trying it at least once

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Let's solve one more math problem

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

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If we fill in a little more information that we know, it is as follows

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First, we can see that the given triangle is isosceles

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because the sum of the internal angles of the triangle is 180 degrees

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Therefore, the length of two sides of the triangle is 5

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Also, if we think of a right triangle with lengths of 3:4:5

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if the base is 5

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we can see that the triangle's height is 3

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Therefore, the area of ​​the triangle is 5 x 3 x 1/2

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which is 15/2

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There is a saying, "There's more than one way to skin a cat"

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You may think that you just have to solve the problem somehow

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Rather than solving a complicated and difficult problem

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it is better to solve the problem easily and simply

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That way, you can reduce calculation errors and solve the problem quickly

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To proceed with the problem-solving process easily and simply

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you must use the information effectively

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To do that, it is important to understand the necessary information

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Let's take a look at a problem to figure out the necessary information

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Question 2. Calculate the area of ​​the shaded right triangle given the following

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Students who learned the Pythagorean theorem in school

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will probably start with complicated calculations

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such as calculating the length of the hypotenuse of a large right triangle

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However, I told you that blindly calculating

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does not lead to good results

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Before calculating, let's take a look at the length ratio

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The length ratio of the simplest right triangle made of natural numbers is

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3:4:5

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Therefore, all triangles with a 3:4:5 ratio

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are right triangles

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The given large right triangle also has two sides of 9 and 12

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This means that 9:12 is 3:4

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Therefore, even without applying the Pythagorean theorem to calculate the hypotenuse

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we can see that it is 9:12:15

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If we solve the problem

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first, we can see that the length of the hypotenuse of the large right triangle is

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9:12:15

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which is 15

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If we write this

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we can see that the length of the hypotenuse of the small right triangle is 10

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If the lengths of the two sides of the small right triangle are a and b

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then the right triangle is also a right triangle with a ratio of 3:4:5

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so a:b:10 is 6:8:10

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Therefore, the area of ​​the small right triangle can be calculated like this

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1/2 x a x b is

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1/2 x 6x 8

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which is 24

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Question 3. Find the size of the given angle x in a regular octagon as follows

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Let's approach this problem in two ways

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The first way is to figure out the given angles without thinking about the idea

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First, let's think about the size of the interior angles of a regular octagon

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We can think of six triangles

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inside a regular octagon as follows

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Therefore, the sum of the interior angles of a regular octagon is 180 degrees x 6

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which is 1080

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Since the sizes of the eight interior angles of a regular octagon are the same

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if we display the size of the angles in the given problem

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we can display all the values ​​of the angles

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If four angles make a square

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45 + 225 + 45 + x is 360

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Therefore, x is 45 degrees

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Let's approach this problem with an idea

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Let's think about the following triangle

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If we think about a triangle that contains angle x like this

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we can see that the two remaining angles

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except x, are angles that divide the interior angles of a regular octagon in half

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Because just as we can think of the center of a circle

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we can think of the center of a regular octagon as a picture

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So in the triangle we drew, the other two angles, excluding x

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are angles that divide the 135 degrees angle of a regular octagon in half

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Therefore, the sum of the two angles is 135 degrees

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and x is 180 degrees minus 135 degrees

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which is the answer of 45 degrees

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Thinking Beyond the Obvious

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One of the charms of Euclidean geometry is that

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you can solve problems by imagining things that are not directly revealed in the problem

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The problems given to us are

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often part of the overall situation

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So, just by drawing extension lines or auxiliary lines

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and expanding the situation a little

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you can often get ideas for solving the problem

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Drawing lines that do not exist

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sometimes copying the shape of the problem

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and moving it to the side

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is an active and positive approach that often gives you ideas for solving the problem

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Let's look at the related content while solving the problem

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Question 4. Find the area of ​​the pentagon ABCDE given below

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The sum of the interior angles of a pentagon is 540 degrees

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This is 180 degrees x 3

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Therefore, if we find the value of angle E

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it is 540 - (90 x 3 + 135)

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which is 135 degrees

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Also, 180-135 equals 45 degrees

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If we understand this relationship

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we can think of two right isosceles triangles

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Since the two sides of an isosceles right triangle are the same length

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we can determine the following information

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The large triangle FGD is also an isosceles right triangle

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Since the side length is 14

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the area of ​​the large isosceles right triangle

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is 14 x 14x 1/2

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which is 98

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and the areas of the two smaller isosceles triangles are

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4 x 4 x 1/2

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which is 8

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6 x 6x 1/2

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is 18

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Therefore, the answer to the area of ​​the pentagon

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is 98 - (8 + 18)

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and the answer is 72

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One thing we need to be careful about is that

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we should not use our imagination

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and do illogical calculations by eye

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For example, some people solve this problem

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by drawing a line connecting B and E

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and thinking that BCDE is a right triangle

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If you set it up like this, it is not logical

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This means ABE is an isosceles right triangle

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AB = AE = 6

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and BE = 10

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This is not logical if you think about the ratio of the lengths of the sides

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of an isosceles right triangle

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Please remember that you cannot solve math problems

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with your logic

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Question 5. What is the size of each x in the following figure?

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So that the rectangle BCDE becomes a square

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Let's take point E

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Then, since angle ADC is 150 degrees

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we can see that angle ADE in triangle ADE is 60 degrees

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and that the lengths of sides AD and DE are the same

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Therefore, triangle ADE is an equilateral triangle

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with all angles at 60 degrees

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Since triangle AED is an equilateral triangle, AE is 2B

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and triangle AEB is an isosceles triangle

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Therefore, angle BAE is angle ABE

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and is 15 degrees

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We are looking for the size of angle x

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to be 60 - 15

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which is 45 degrees

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If we express it as calculating and imagining

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we should be good at calculating and imagining

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If we only focus on calculating quickly

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and think of the process of imagining leisurely as a waste of time

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it will be difficult to learn truly fun math

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Our math skills will not improve much either

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Our daily lives are the same

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It is necessary to do things quickly and efficiently

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but it is also necessary to think bigger and imagine new things by looking around

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If you are working hard but are not getting much better than average results

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you need to take your time and imagine more

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rather than blindly making plans to do things faster

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Question 6. Find the area of ​​the given rectangle ABCD

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Let's solve this problem

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How would you solve it?

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It seems like we could solve this problem by quickly calculating

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However, since we haven't learned much about calculation methods in this course

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let's solve this problem by imagining

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Let's copy the given shape and paste it

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If we paste four of the same shapes like this

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we can see that it becomes a square with a side length of 5

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Therefore, the square area is 25

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and since we pasted four shapes

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the area of ​​one shape we are looking for is 4/25

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Some people like to paste, and some people like to divide

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If you like to divide

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draw a line segment BD

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Then, cut triangle ABD

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and paste it so that AD becomes CD

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Since angle A plus angle C is 180 degrees

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DBB' becomes a triangle

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and B' becomes an isosceles right triangle, which is B'D

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All we need now is to find the area

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of ​​an isosceles right triangle whose hypotenuse is of length 5

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The area of ​​an isosceles right triangle

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can be calculated by attaching four of the same ones

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If we attach the four isosceles right triangles

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we will create a square with a side length of 5

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Therefore, the area of ​​the square is 25

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and the area of ​​one isosceles right triangle that we are looking for

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is 25/4

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In other words, the area of ​​the shape we want to calculate is 25/4

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We see that difficult problems can be solved

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by simply examining them logically one by one while using various imaginations

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without doing difficult calculations

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This is how we experience the fun of math

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Question 7. In the following figure, find the area of ​​the shaded area when AB is 6

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Let's approach this problem not by calculating but by imagining

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Let's think of the given shapes as tiles on the floor

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If we think of them as tiles on the floor

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we can attach them, as you can see on the screen now

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If we attach four shapes like this

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we can see a square with a side length of 6 is created

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The area of ​​the given shape is 6 x 6

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A quarter of a square with a side length of 6

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so the answer is 9

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his problem can also be solved using another analytical method

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Since solving one problem in various ways is a good way to study

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let's solve it using another method

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In this problem, the area of ​​the shaded area

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can be thought of as a square and a right triangle

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It can be interpreted as the area of ​​3/2 + 1

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5/2 squares

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Therefore, if you find the area of ​​one square with the given condition

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you can find the area of ​​the shaded area

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by finding 5/2 of it

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The area of ​​one square can be found as follows

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The area of ​​the shaded triangle is 3/2

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If you attach the shape on the left as on the right

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you will create a large square with a side length of 6

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which has 4 shaded triangles and 4 squares

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Therefore, if we call the area of ​​one square A

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we can set up the following equation

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If we follow the equation you just saw, A will be 18/5

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We said that the area of ​​the shaded area

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is 5/2 times A

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Since the area of ​​a square of the value of A is 18/5

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the answer to 5/2a is 5/2 x 18/5

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which is 9

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I solved the problem by imagining rather than calculating

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After learning the Pythagorean theorem

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and solving quadratic equations in middle school

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many problems are approached by calculating

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Calculating this way can solve problems more easily

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but sometimes, you will experience such calculations blocking your imagination

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Since this course aims to solve all problems

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with 4th-grade knowledge

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I solved the problem by imagining rather than calculating

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I hope this can provide insight to you, the digital content creators

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Rather than quickly coming up with an answer

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you can think of something else with various imaginations

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Converting given information into necessary information
The more information you have, the easier and faster you can approach problem-solving

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It is an effective way to approach the solution by displaying information that is likely to be used in problem-solving
To proceed with the problem-solving process easily and simply, you must use the given information effectively

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Imagine a wider part
The problems given to us are often part of the overall situation

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We can get an idea for solving the problem just by drawing an extension line or auxiliary line and expanding the situation a little
However, we should not use our imagination and make illogical calculations by guessing

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Even without making difficult calculations, we can solve difficult problems by exercising our imagination and logically considering them