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
4 Finding Ideas with Euclidean Thinking 2 Thinking in Parts

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Hello, this is Jongha Park

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In this session, we'll explore a frequently used key thinking technique

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analysis

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Analysis is the process of breaking down something unfamiliar into parts

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we are already familiar with

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It involves dividing complex and challenging problems into smaller, manageable ones

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that are easier to understand and handle

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A smart person isn't simply tackling complexity head-on

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but skillfully breaking down complexity into simpler, more approachable components

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This method is likely very familiar to those of you creating digital content

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Let's dive into the analysis concept by solving problems in Euclidean geometry

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Divide and Simplify

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Many problems can be easily solved simply by dividing them into parts

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This is often the case with Euclidean geometry problems as well

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When approaching a problem

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try breaking it down into simpler, more understandable components

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Keep the divisions straightforward

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like drawing horizontal and vertical lines

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Let’s take a look at an example and see how this works

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Question 1. What is the area of the shaded region in the diagram?

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To analyze the problem, let’s draw an auxiliary line as shown

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Since the area of a triangle is calculated as half the product of its base and height

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triangles with the same base and height will have the same area

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By dividing the diagram, we can identify 8 triangles

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grouped into two pairs of triangles with equal areas

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From the problem’s conditions, we can deduce the following relationships

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a+d+b+c=

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7+5=

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12

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Also, c+d=8

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Using these relationships

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we can calculate the value of a+b

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If we put c+8 at the front equation

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a+b=4

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Question 2. The given quadrilateral divides the area into four triangles

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with areas 9, 12, 23, and x

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Find the value of x

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I think this problem requires dividing a triangle

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and setting up a relationship to calculate it

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Before we do the calculations, let's think about it this way

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First, if we divide the rectangle into four parts

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by drawing lines as shown below

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the areas of the two triangles with areas of 9 and 23

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divide the area of ​​the entire rectangle exactly in half

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Similarly, the triangles with areas 12x

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also sum up to half the total area

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Do you see the relationship?

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So, 9+23=12+x

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x=20

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Through this problem, we can conclude the following

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When the interior of a square is divided into four triangles

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the areas of triangles A, B, C, and D have the following relationship

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A + B = C + D =

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half the area of the triangle

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It is not the right way to formalize and memorize this relationship

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It is necessary to fully understand the related content

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and easily apply it to solve the problem

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when it is presented as a problem

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If you try to memorize even such trivial things

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your head may explode because there is too much to memorize

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You can solve problems more effectively

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by understanding and applying them rather than memorizing them

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People who are good at solving complex problems

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don't have ability handling complex things well

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Their ability is mainly

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demonstrated by dividing complex issues into several simple problems

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Dividing them into small parts and expressing them simply

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is a very effective way to solve problems

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Let's check it out through the question

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Question 3. In the following figure, when AC is 6, and BC is 6

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find the area of ​​the shaded

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To find the area of ​​the shaded

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let's divide it as follows

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If we divide it like this, we can think of the whole thing as 4 triangles

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If we only consider triangle ACD

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triangle ACD has three small triangles

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AEF, ECF, and CDF

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ECF and CDF are symmetrical

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so they are congruent and have the same area

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Triangles AEF and ECF have the same height

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and their bases are 2 and 4

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ratio is 1:2

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Therefore, the ratio of the areas of triangles AEF and ECF is 1:2

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In conclusion, the ratio of the areas of the three small triangles

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AEF, ECF, and CDF, inside the large triangle ACD

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is 1:2:2

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The area of ​​ACD is 4 x 6 x 1/2 =

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12

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Therefore, the AEF and ECF CDF areas are determined as

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12/5, 24/5, and 24/5

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It's like this

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Therefore, the area of ​​the shaded area, ECF + CDF is

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24/5 + 24/5

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So, the answer is 48/5

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Let's continue solving other question

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Question 4. Find the area of ​​the shaded area below

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To approach this problem similarly

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let's divide the shaded area into two parts, as shown

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Based on the horizontal line of the large triangle

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the two triangles with bases of 6 and 3 have the same height

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so the ratio of their areas is 2:1

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Therefore, let's write the area as 2A, A

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In the same way, based on the vertical line of the large triangle

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the two triangles with bases of 4 and 8

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have the ratio of their areas 1:2

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Let's write this area

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as B,2B

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Specifically

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the area of B + 2B + 2A is

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6 x 12 x 1/2 =

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36

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And the area of 2B + 2A + A is

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8 x 9 x 1/2 =

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36

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This can be expressed as a formula as follows

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2A + 3B = 36

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3A + 2B = 36

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In conclusion, 5A + 5B = 72

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and A + B+ 72/5

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Since the area of the shaded is

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2A + 2B

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So, the answer is 144/5

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Analytical Tools

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This time, the problems we are dealing with are

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simple figures, triangles, and squares

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It would be helpful to organize what you know about

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triangles and squares to solve the problem

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We often apply the area of ​​a triangle

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The area of ​​a triangle can be effectively used

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by using the following theorem: the area is the same if the base

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and height are the same

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The word tool is used to express frequently used things

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When solving problems about the area

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it is good to think of the following theorem about the area

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of ​​a triangle as a useful analysis tool

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Let's look at it again while solving the question

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Question 5. Find the area of ​​part A minus the area

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of ​​part B in the following figure

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Before solving this problem, let's think about this

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First, let's look at the following figure

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Triangles ABC and ABD

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have the same base and height

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so their areas are the same

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However, the circled area is the common part of the two triangles

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Therefore, the areas of S and S' are the same

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This relationship is often used in the content

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about triangles

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and it is like a sense gained through experience

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The way to become wise is to gain this sense

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through problem-solving experience

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Now, let's apply this to problem 5

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If you draw a line like the following in the given problem

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the areas of B and A1 are the same

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This is because B and A1 are

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the common parts of two triangles with the same base and height

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Therefore, you can think that their areas are the same

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When you subtract the area of ​​part B from the area of ​​part A

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you get a remainder of a triangle with a base of 6 and a height of 2

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That is, 2 x 6 x 1/2 =

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6

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Question 6. Given a rectangle ABCD as shown

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find the area of ​​the hatched part

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The area of ​​a rectangle is the product of its length and width

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However, it is not easy to find the area of ​​an irregular rectangle

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If we think of special rectangles

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we can think of parallelograms and trapezoids

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The area of ​​a parallelogram and a trapezoid is calculated as follows

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This can be easily understood by dividing the parallelogram

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and trapezoid into triangles

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If we solve Problem 6, the area of ​​rectangle ABCD

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can be found as follows

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It is 1/2 x (2 + 5) x 4 =

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14

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The strategy for solving this problem is to

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divide the shape given in the problem as shown below and subtract the areas

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of s1, s2, s3, and s4 from the whole to find the area of ​​the shaded part

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First, let's consider triangle ABE

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with line segment AB as the center

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The area of ​​triangle ABE

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can be found by subtracting

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the areas of triangle ADE and triangle BCE from the entire rectangle ABCD

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Therefore, the area of ​​triangle ABE is

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14 - (6/2 + 5/2) =

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17/2

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The ratio of the area of ​​triangle BEG to triangle ABE is

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2/5

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Therefore, the area of ​​triangle BEG is 17/2 x 2/5 =

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17/5

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This time, let's consider triangle ABF

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centered on line segment AB

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The area of ​​triangle ABF can be found by subtracting the areas

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of triangles ADF and BCF from the entire quadrilateral ABCD

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Therefore, the area of ​​triangle ABF is

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14 - (2 + 5) = 7

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The ratio of the area of ​​triangle AFH to triangle ABF is 1/5

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Therefore, the area of ​​triangle AFH is 7/5

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If we write the information about the area we know

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on the given figure, it is as follows

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Therefore, the area of ​​the hatched part we are finding

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is 14 - (2 + 7/5 + 17/5 + 5/2) =

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47/10

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This problem was very difficult

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Not many people would have solved it easily

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The reason this problem is difficult is not because it requires difficult concepts or knowledge

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It is because it is a problem that requires thinking about a complex problem step by step in a simple form

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People who do not have much experience with such problems find it difficult

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It is good to remember once again that complex problems need to be approached

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by breaking them down into simpler forms

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Let's look at another problem that requires ideas rather than knowledge to solve the question

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Question 7. In the following figure, the numbers in the squares represent areas

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Find the area of ​​the shaded

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First, let's think about a square with an area of ​​43

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by dividing it into areas

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of 21 + 22 = 43

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Since there is also a square on the far left with an area of ​​21

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we can think of the following lengths

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Now, if we look at the square at the bottom

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we can think of a square with an area of ​​4 x 10, which is 40

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and the area of ​​the square to its right

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is 40 - 29

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= 11

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If we compare the squares with areas 11 and 22

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we can see that the length of one side of the square on the left is 8

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Therefore, the area of ​​the square on the left is 80

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and the area of ​​the shaded region is 80 - (21 + 22)

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= 37

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I introduced a problem with a very low correct answer rate

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The reason why this problem has a low correct answer rate

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is not because it requires deep knowledge or concepts

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It is a problem that even elementary school students can easily answer

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but it is a problem that requires problem-solving experience that approaches the problem in various ways

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Math problems require free thinking rather than relying on formulas

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Problem-solving experience is required, and I think various problem-solving experiences

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are important

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This also applies to you, the digital content creators

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Having more knowledge does not mean producing better content

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Rather than knowledge, I think it is necessary

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to have various experiences and extract the insights you need from those experiences

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I hope you will use the analysis you experienced today in complex situations

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Divide a large problem into smaller ones that are easy to handle

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Remember that most smart people's abilities

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start from dividing a complex problem into several simple problems

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and I hope you will wisely apply this to your problems

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Divide and simplify
A process is needed to fully understand the content and easily apply it to solve problems

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Problems can be solved more effectively by understanding and applying it rather than memorizing it
Dividing it into small parts and expressing it simply is an effective way to solve problems

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Analytical Tools
Before solving a problem, it is also helpful to organize what you know

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Formulas related to shapes are obtained through experience, and through the experience of solving problems, you can gain a sense of the solution

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The word tool is used to express frequently used content, and analysis tools are useful when solving problems
Math problems should be approached with free thinking rather than relying on formulas

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You can gain the insight you need through the experience of solving problems by approaching them in various ways