Euclidean Geometry
Logic and Problem Solving
Euclidean Logical Thinking for Digital Content Creators 7. Problem-Solving Techniques to Break Free from Conventional Answers Part 2
Hello, this is Jongha Park
In this session, we will approach problems from a new perspective
introducing a problem-solving technique that involves actively
transforming problems
Everyone has their own framework of thought
We rely on common sense and think within certain boundaries
which helps us navigate everyday life efficiently
However, as times change and circumstances evolve
past conventions often fail to apply today
What was right yesterday may be wrong today
and what was wrong yesterday might be right today
We live in a world where such shifts are increasingly common
In such a world, flexibility of thought and freedom from stereotypes
are essential
For you, digital content creators
it’s crucial to propose new thought frameworks
and guide your content consumers
This session will help you experience and learn techniques
for breaking free from conventional thinking
Approaching Problems from a New Perspective
Let’s try a quiz
In the matchstick equation given below
move only one matchstick to make the equation correct
There are three solutions
Try finding all three
This quiz is not related to math questions
but it helps broaden your thinking and develop techniques
for solving math problems
Approach it with flexible thinking
and try to find all three solutions
Pause the video, solve the problem
and then resume to check your answers
The answers are as follows
Have you confirmed the solutions?
Now, with this flexible thinking, let’s solve a Euclidean geometry problem
Problem 1: Calculate the combined area of the two shaded triangles in the figure below
This problem becomes straightforward once you understand the situation
However, without flexible thinking, finding the answer can be very challenging
Here’s how you can understand the problem
The area of a triangle is base x height x 1/2
Triangles with the same base and height
have the same area, regardless of their shape
Now, in the given figure, identify triangles
with the same area as the shaded ones
For example, the red triangle and the shaded triangle have the same area
because they share the same base and height
The two red triangles perfectly divide the rectangle into halves
Thus, the combined area of the red triangles is
5 x 8 x 1/2
= 20
This is also the combined area of the shaded triangles
Solving one problem
allows us to gain experience with related scenarios
Just like gaining experience points in a game helps you level up
these problem-solving experiences enhance your abilities
Let’s use what we learned from Problem 1 to solve another problem
Problem 2: In the figure below, there is a shaded triangle within the hexagon
The total area of the hexagon is 24
What is the area of the shaded triangle?
Begin by examining what proportion of the hexagon
the shaded triangle occupies
The shaded part lies entirely in
the upper half of the hexagon
Let's look at the upper part first
We are looking at the colored triangle
But let’s shift focus to the unshaded triangle
Instead of seeing it as
a white background with a blue triangle
consider it as a blue background with a white triangle
If you look at the white triangle
you can also think of it as the following triangle, as they share the same base and height
By looking it like this, we can see that the white triangle
is 1/3 of the upper half of the hexagon
Therefore, the shaded triangle
constitutes 2/3 of the upper half
or 2/6 of the entire hexagon
Given that the hexagon’s total area is 24, the shaded triangle’s area is 8
When solving math problems, there is something called a "model answer"
It refers to the method taught by teachers
or presented in textbooks
One thing to keep in mind is that the model answer is not the only correct answer in the world
It's better to solve math problems using various methods
The more ways you solve problems, the more your math skills improve
So, instead of stopping at the model answer
it’s good to solve problems in various ways, including the model answer
If you're taking a math test, you'll need to find the answer quickly using a single method
However, when studying math, try solving a single problem in multiple ways
The more ways you attempt
the more effective your learning will be
Try the model solution, and also experiment with unconventional methods
In particular, if you frequently use what may seem like shortcuts or tricks, you'll develop a sense of intuition
Mathematical intuition is very important
Let’s examine this concept through a problem
Problem 3: In the given figure, calculate the area of the shaded triangle
Some people say they solved this problem in 5 seconds
They didn’t just calculate randomly
they came up with an insightful idea
Let’s solve this problem using two methods
First, let’s approach it with the model solution
This problem requires using the fact that the two right triangles, ABC and EDC, are similar
By identifying corresponding sides, we can set up a proportion
AB/BC = ED/DC
From this, we can perform the following calculations
Ultimately, BC × ED = 6 × 4
The area of triangle BCE is half the product of the lengths of BC and ED
Therefore, the area of triangle BCE is 1/2 x 6 x 4
= 12
But let’s think about this differently
As explained in the previous session
the areas of the shaded region (S) and S’ are equal
Looking at the figure
this is because the remaining parts of two triangles
with equal areas are also equal
With this thought, let’s reexamine the problem
In the given problem, draw a line connecting AD
to create triangle ACD
Now compare triangles ABE and ABD
Both triangles share the same base, AB
and since AB is parallel to DE, their heights are the same
Thus, the areas of triangles ABE and ABD are equal
If you subtract the common area ABC from these two equal-area triangles
you can conclude that the areas of triangles BCE and ACD are the same
Triangle ACD has a base of 4 and a height of 6
Thus, the area of triangle ACD is 1/2 x 4 x 6
= 12
which is the same as the area of triangle BCE
Let’s continue solving problems
Problem 4: Calculate the area of the shaded right quadrilateral in the given figure
This problem can also be approached in two ways
For a quick calculation
use the similarity between triangles ADF and FEC
From the similarity ratio, you can find the product of DF and FE
Since the area of a right quadrilateral is simply the product of its base and height
DF × FE gives the area of the quadrilateral
The similarity ratio is calculated as follows
The area of right quadrilateral DBEF is 36
But let’s look at the problem differently
The given right quadrilateral
can be thought of as a part created by the diagonal of a rectangle
Thinking this way
A and B are equal, and C and D are also equal
Thus, the areas of the two shaded regions are equal
The area of the rectangle marked in red is 3 x 12
which is 36
so the area of the quadrilateral we’re looking for is also 36
When solving Problems 3 and 4
some people claim they can find the answers
within 3 seconds
If we solve the problems using what we call “shortcuts"
it might indeed be possible to find the answers in such a short time
While I’ve referred to these methods as the model answer and the shortcut
thinking in terms of shortcuts is important as well
It reflects a deep understanding of the problem
and brings innovative ideas to problem-solving
The key is to understand the problem from various perspectives
and to develop a sense for solving problems
One thing to be cautious about
A “shortcut” should not mean a guess made
without logical rigor
or clear reasoning
Even when using shortcuts
you must maintain logical thoroughness and base your reasoning on solid grounds
Shortcuts do not mean guessing blindly
they mean finding a quicker path than usual
through intuition and sense
For digital content creators
demonstrating a sense of intuition while working is even more critical
Can there be a model answer when creating something new?
True creation is implemented through
the creator’s sense of intuition
Remember the importance of intuition
Let me introduce another problem
Problem 5: Find the area of the square in the following diagram
This problem requires setting variables and performing calculations
You’ll need to apply concepts like the Pythagorean theorem from the textbook to solve it
However, before diving into complex calculations
let’s solve it using the following method
Divide the given figure like this
If divided this way, the entire right isosceles triangle is split
into nine smaller right isosceles triangles that are identical in shape and size
The square we want to find the area of consists of four out of these nine parts
Thus, the area of the square is
4/9 of the total area
The area of the larger right isosceles triangle is
10 x 10 x 1/2
=50
So 50 x 4/9
200/9 is the area of the square
Actively Engaging with Problems
When solving problems, instead of passively waiting for ideas to come to you
you should actively engage with the problem
For instance, in Euclidean geometry
you might extend lines, draw auxiliary lines
or even cut and attach sections of a shape to other parts
Such active efforts are essential for problem-solving
Sometimes, merely imagining cutting out
and attaching a certain part to a suitable location
can lead to a solution
When a problem is solved in this way, it often brings an exhilarating sense of achievement
During practice, try drawing lines or cutting and attaching shapes on paper
to solve a problem or two
By cutting and attaching paper while solving problems
you can expand your perspective
and enhance your imagination
Problem 6: In the following scenario, calculate the area of triangle ABC
A methodical approach would be
to use the similarity of triangles and calculate the similarity ratio
But let’s consider it differently
Let’s think about triangle CDE
Triangle CDE has the same area as triangle ACD because
they share the same base and height
By drawing and observing triangle CDE
we find that the area of triangle ABC
is the same as the area of triangle BDE
Since BE = 10 and CD = 4
the area of triangle BDE is 10 x 4 x 1/2
= 20
Thus, the area of triangle ABC is also 20
Problem 7: Calculate the area of quadrilateral ABCD as given in the following conditions
Among the problem's conditions, note that
AM = BM
Looking at M as the center AMD+BMC=90 degrees
Thus, by cutting triangle AMD
and attaching it to make AM equal to BM
the entire figure forms a new triangle as shown below
When cut and reattached this way the area of the quadrilateral we’re solving for
becomes the area of triangle DCD′
the area of triangle DCD′ is 12 x 8 x 1/2
= 48
Let's continue
Problem 8: In the following diagram, AB = AD
D + B=180º
AC = 10
and ACB=15º
Find the area of quadrilateral ABCD
Given the condition AB = AD
and D + B = 180º
by cutting triangle ACD and attaching it so that points B and D align
we find that segments BC and DC fall on the same straight line
When represented visually, it appears as shown below
Since AC has been cut and reattached, AC becomes AC'
Because it is an isosceles triangle, angle AC'B=15º
Now, our task is to calculate the area of the isosceles triangle ACC'
To find the area of triangle ACC'
consider the right triangle on a continuous line as follows
Since AC = 10, AC'=10 and EC'=5
The area of triangle ACC′ can be calculated as the following
1/2 x 2C' x AC = 25
= 25
By cutting, reattaching, and approaching the problem
in various ways
the moment a problem is solved brings a significant sense of accomplishment
This sense of accomplishment stems from actively engaging with the problem
It’s a feeling similar to the joy experienced by creators
when they make something new
Rather than passively fitting into a predefined mold
all of your digital content creators
should actively creating a new framework
We study math not just to learn concepts and knowledge
but more importantly, to develop problem-solving skills
Everyone learns through experience
Sometimes people tease those
inexperienced in relationships by saying
I learned about love through books
While reading and acquiring knowledge is important
learning through experience is even more valuable
In this session, we’ve tackled many problems
Through these experiences
apply the problem-solving techniques you’ve learned to your own challenges
As digital content creators
we’ve explored Euclidean geometry across seven sessions
As emphasized at the beginning, the most important thing is fun
Did you have fun?
By experiencing cognitive enjoyment, we hope you’ve not only enjoyed math and logical thinking
but also naturally improved your problem-solving skills
Apply these thinking techniques to your creative activities
and achieve even better results
Always remember: fun and confidence are key
We cheer for you as digital content creators
Approaching problems from a new perspective Exploring problems through many methods enhances skills so solving through diverse approaches is important
Trying various methods can enhance your problem-solving skills You must create intuition for solving problems
Even if you use a shortcut, it should be backed by logical and solid reasoning
Actively Engaging with Problems Actively engaging with the problem is crucial, such as drawing extended or auxiliary lines, and cutting shapes
The moment you solve a problem after approaching it in different ways, you feel a significant sense of achievement
Rather than passively conforming to pre-existing frameworks, creating new frameworks actively leads to a deeper sense of fulfillment