Euclidean Geometry
Logic and Problem-Solving
Logical Thinking in Euclidean Geometry for Digital Content Creators 4 Finding Ideas with Euclidean Thinking 2 Thinking in Parts
Hello, this is Jongha Park
In this session, we'll explore a frequently used key thinking technique
analysis
Analysis is the process of breaking down something unfamiliar into parts
we are already familiar with
It involves dividing complex and challenging problems into smaller, manageable ones
that are easier to understand and handle
A smart person isn't simply tackling complexity head-on
but skillfully breaking down complexity into simpler, more approachable components
This method is likely very familiar to those of you creating digital content
Let's dive into the analysis concept by solving problems in Euclidean geometry
Divide and Simplify
Many problems can be easily solved simply by dividing them into parts
This is often the case with Euclidean geometry problems as well
When approaching a problem
try breaking it down into simpler, more understandable components
Keep the divisions straightforward
like drawing horizontal and vertical lines
Let’s take a look at an example and see how this works
Question 1. What is the area of the shaded region in the diagram?
To analyze the problem, let’s draw an auxiliary line as shown
Since the area of a triangle is calculated as half the product of its base and height
triangles with the same base and height will have the same area
By dividing the diagram, we can identify 8 triangles
grouped into two pairs of triangles with equal areas
From the problem’s conditions, we can deduce the following relationships
a+d+b+c=
7+5=
12
Also, c+d=8
Using these relationships
we can calculate the value of a+b
If we put c+8 at the front equation
a+b=4
Question 2. The given quadrilateral divides the area into four triangles
with areas 9, 12, 23, and x
Find the value of x
I think this problem requires dividing a triangle
and setting up a relationship to calculate it
Before we do the calculations, let's think about it this way
First, if we divide the rectangle into four parts
by drawing lines as shown below
the areas of the two triangles with areas of 9 and 23
divide the area of ​​the entire rectangle exactly in half
Similarly, the triangles with areas 12x
also sum up to half the total area
Do you see the relationship?
So, 9+23=12+x
x=20
Through this problem, we can conclude the following
When the interior of a square is divided into four triangles
the areas of triangles A, B, C, and D have the following relationship
A + B = C + D =
half the area of the triangle
It is not the right way to formalize and memorize this relationship
It is necessary to fully understand the related content
and easily apply it to solve the problem
when it is presented as a problem
If you try to memorize even such trivial things
your head may explode because there is too much to memorize
You can solve problems more effectively
by understanding and applying them rather than memorizing them
People who are good at solving complex problems
don't have ability handling complex things well
Their ability is mainly
demonstrated by dividing complex issues into several simple problems
Dividing them into small parts and expressing them simply
is a very effective way to solve problems
Let's check it out through the question
Question 3. In the following figure, when AC is 6, and BC is 6
find the area of ​​the shaded
To find the area of ​​the shaded
let's divide it as follows
If we divide it like this, we can think of the whole thing as 4 triangles
If we only consider triangle ACD
triangle ACD has three small triangles
AEF, ECF, and CDF
ECF and CDF are symmetrical
so they are congruent and have the same area
Triangles AEF and ECF have the same height
and their bases are 2 and 4
ratio is 1:2
Therefore, the ratio of the areas of triangles AEF and ECF is 1:2
In conclusion, the ratio of the areas of the three small triangles
AEF, ECF, and CDF, inside the large triangle ACD
is 1:2:2
The area of ​​ACD is 4 x 6 x 1/2 =
12
Therefore, the AEF and ECF CDF areas are determined as
12/5, 24/5, and 24/5
It's like this
Therefore, the area of ​​the shaded area, ECF + CDF is
24/5 + 24/5
So, the answer is 48/5
Let's continue solving other question
Question 4. Find the area of ​​the shaded area below
To approach this problem similarly
let's divide the shaded area into two parts, as shown
Based on the horizontal line of the large triangle
the two triangles with bases of 6 and 3 have the same height
so the ratio of their areas is 2:1
Therefore, let's write the area as 2A, A
In the same way, based on the vertical line of the large triangle
the two triangles with bases of 4 and 8
have the ratio of their areas 1:2
Let's write this area
as B,2B
Specifically
the area of B + 2B + 2A is
6 x 12 x 1/2 =
36
And the area of 2B + 2A + A is
8 x 9 x 1/2 =
36
This can be expressed as a formula as follows
2A + 3B = 36
3A + 2B = 36
In conclusion, 5A + 5B = 72
and A + B+ 72/5
Since the area of the shaded is
2A + 2B
So, the answer is 144/5
Analytical Tools
This time, the problems we are dealing with are
simple figures, triangles, and squares
It would be helpful to organize what you know about
triangles and squares to solve the problem
We often apply the area of ​​a triangle
The area of ​​a triangle can be effectively used
by using the following theorem: the area is the same if the base
and height are the same
The word tool is used to express frequently used things
When solving problems about the area
it is good to think of the following theorem about the area
of ​​a triangle as a useful analysis tool
Let's look at it again while solving the question
Question 5. Find the area of ​​part A minus the area
of ​​part B in the following figure
Before solving this problem, let's think about this
First, let's look at the following figure
Triangles ABC and ABD
have the same base and height
so their areas are the same
However, the circled area is the common part of the two triangles
Therefore, the areas of S and S' are the same
This relationship is often used in the content
about triangles
and it is like a sense gained through experience
The way to become wise is to gain this sense
through problem-solving experience
Now, let's apply this to problem 5
If you draw a line like the following in the given problem
the areas of B and A1 are the same
This is because B and A1 are
the common parts of two triangles with the same base and height
Therefore, you can think that their areas are the same
When you subtract the area of ​​part B from the area of ​​part A
you get a remainder of a triangle with a base of 6 and a height of 2
That is, 2 x 6 x 1/2 =
6
Question 6. Given a rectangle ABCD as shown
find the area of ​​the hatched part
The area of ​​a rectangle is the product of its length and width
However, it is not easy to find the area of ​​an irregular rectangle
If we think of special rectangles
we can think of parallelograms and trapezoids
The area of ​​a parallelogram and a trapezoid is calculated as follows
This can be easily understood by dividing the parallelogram
and trapezoid into triangles
If we solve Problem 6, the area of ​​rectangle ABCD
can be found as follows
It is 1/2 x (2 + 5) x 4 =
14
The strategy for solving this problem is to
divide the shape given in the problem as shown below and subtract the areas
of s1, s2, s3, and s4 from the whole to find the area of ​​the shaded part
First, let's consider triangle ABE
with line segment AB as the center
The area of ​​triangle ABE
can be found by subtracting
the areas of triangle ADE and triangle BCE from the entire rectangle ABCD
Therefore, the area of ​​triangle ABE is
14 - (6/2 + 5/2) =
17/2
The ratio of the area of ​​triangle BEG to triangle ABE is
2/5
Therefore, the area of ​​triangle BEG is 17/2 x 2/5 =
17/5
This time, let's consider triangle ABF
centered on line segment AB
The area of ​​triangle ABF can be found by subtracting the areas
of triangles ADF and BCF from the entire quadrilateral ABCD
Therefore, the area of ​​triangle ABF is
14 - (2 + 5) = 7
The ratio of the area of ​​triangle AFH to triangle ABF is 1/5
Therefore, the area of ​​triangle AFH is 7/5
If we write the information about the area we know
on the given figure, it is as follows
Therefore, the area of ​​the hatched part we are finding
is 14 - (2 + 7/5 + 17/5 + 5/2) =
47/10
This problem was very difficult
Not many people would have solved it easily
The reason this problem is difficult is not because it requires difficult concepts or knowledge
It is because it is a problem that requires thinking about a complex problem step by step in a simple form
People who do not have much experience with such problems find it difficult
It is good to remember once again that complex problems need to be approached
by breaking them down into simpler forms
Let's look at another problem that requires ideas rather than knowledge to solve the question
Question 7. In the following figure, the numbers in the squares represent areas
Find the area of ​​the shaded
First, let's think about a square with an area of ​​43
by dividing it into areas
of 21 + 22 = 43
Since there is also a square on the far left with an area of ​​21
we can think of the following lengths
Now, if we look at the square at the bottom
we can think of a square with an area of ​​4 x 10, which is 40
and the area of ​​the square to its right
is 40 - 29
= 11
If we compare the squares with areas 11 and 22
we can see that the length of one side of the square on the left is 8
Therefore, the area of ​​the square on the left is 80
and the area of ​​the shaded region is 80 - (21 + 22)
= 37
I introduced a problem with a very low correct answer rate
The reason why this problem has a low correct answer rate
is not because it requires deep knowledge or concepts
It is a problem that even elementary school students can easily answer
but it is a problem that requires problem-solving experience that approaches the problem in various ways
Math problems require free thinking rather than relying on formulas
Problem-solving experience is required, and I think various problem-solving experiences
are important
This also applies to you, the digital content creators
Having more knowledge does not mean producing better content
Rather than knowledge, I think it is necessary
to have various experiences and extract the insights you need from those experiences
I hope you will use the analysis you experienced today in complex situations
Divide a large problem into smaller ones that are easy to handle
Remember that most smart people's abilities
start from dividing a complex problem into several simple problems
and I hope you will wisely apply this to your problems
Divide and simplify A process is needed to fully understand the content and easily apply it to solve problems
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
Analytical Tools Before solving a problem, it is also helpful to organize what you know
Formulas related to shapes are obtained through experience, and through the experience of solving problems, you can gain a sense of the solution
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
You can gain the insight you need through the experience of solving problems by approaching them in various ways