Euclidean Geometry
Logic and Problem-Solving
Logical Thinking in Euclidean Geometry for Digital Content Creators 5 Finding Ideas with Euclidean Thinking 3. Identifying Known Shapes
Hello, this is Park Jongha
Today, let's look at the most important keyword
in problem-solving: connection
What is studying?
It's learning what you don't know
Learning what you don't know means
understanding and remembering what you don't know
by connecting it to what you already know
If you understand the relationship with something you know well and are familiar with
you will understand what you don't know and remember it for a long time
However, if you don't connect what you don't know to what you know
and just try to accept it
you won't understand it, and it will be difficult to remember for a long time
This is logic
Logic is understanding what you don't know with what you know
Persuasion is connecting what the other person knows
with what you are arguing
Problem-solving is the same
The only way to solve a problem you don't know is
to connect what you know with what you don't know
This is something that is necessary for you, the digital content creators
and this time, we will look at this content
Finding Familiar Elements
I emphasized that problem-solving is finding the answer
by connecting what you don't know with what you know
So when you come across a problem that you don't know well and are unfamiliar with
you should look for something you already know well and are familiar with
Find something you know and figure out the relationship with what you don't know
That's because, through that process, you find the answer to the problem you don't know
Let's take a closer look at the problem
Question 1. Find the area of ​​the given triangle
To solve this problem, let's think about the familiar
30 degrees, 60 degrees, and 90 degrees right triangles
Since the given angle in the problem is 30 degrees
let's draw a 30 degrees, 60 degrees, and 90 degrees right triangle as shown below
30 degrees, 60 degrees, and 90 degrees right triangles
are made from equilateral triangles
If we divide an equilateral triangle in half
we get 30 degrees, 60 degrees, and 90 degrees right triangles
The thing to note here is the ratio of the lengths
The length of the hypotenuse of a right triangle is twice the length of the shorter side
If we apply the ratio of the lengths of 30 degrees, 60 degrees, and 90 degrees right triangles
to the situation in the problem we looked at earlier
we can see that the height of the given triangle is 4
because the hypotenuse is 8
To summarize, the triangle in the question
has a base of 6 and a height of 4
Therefore, the area of ​​the triangle is 6 x 4 x 1/2
which is 12
The ratios of the lengths of the 30 degrees, 60 degrees, and 90 degrees right triangles made by cutting an equilateral triangle in half
are crucial when solving Euclidean
If we apply the Pythagorean theorem, we can see that the ratio of the lengths of these triangles
is 1 : root of 3 : 2
but since we are not dealing with the Pythagorean theorem
we'll not use the 1 : root of 3 : 2 ratio
and will only use the fact that
the ratio of the lengths of the hypotenuse and the shorter side is 2 to 1, as we have just seen
Another right triangle that is frequently used
is the 45 degrees, 45 degrees, and 90 degrees right triangles
which are made by cutting
a square in half diagonally
The ratio of the lengths of these right triangles
is often used to mean that the lengths of the two sides, except the hypotenuse, are equal
It would be good to look at how these right triangles are used
while solving problems
Question 2. Find the value of x in the given figure below
This problem is complicated
but let's think about the original pieces that were cut off
Imagine the pieces that might have been cut off
So, let's try to attach the pieces that might have been cut off
The angles of A and E are 120 degrees
so 180 degrees minus 120 degrees is 60 degrees
Thinking about this, we can think about
attaching the following equilateral triangles in addition to AE
After attaching this equilateral triangle to AE
we can attach right triangles with 30 degrees, 60 degrees, and 90 degrees to the remaining BC
to make the entire thing one large 30 degrees, 60 degrees, and 90 degrees right triangle
Since the entire large right triangle GDF is a 30 degrees, 60 degrees, and 90 degrees right triangle
the ratio of the hypotenuse length to the shorter side is 2:1
DF is 5 + 7
which is 12
so GF is 24
Since GB is 6 and AF is 7
GF is GB plus x plus AF
which is 6 + x + 7
which is 24
Therefore, x is 11
Let's look at another question
Question 3. Given the figure below, find the length of AB
If you imagine this problem is similar to Problem 2 with one part cut off
and think about the original shape before it was cut off
you can think of an equilateral triangle OBC, as you can see now
The cut-off part is
a right triangle OAD with angles of 30, 60, and 90 degrees
Since OBC is an equilateral triangle
OD is 2
and the right triangle OAD is a right triangle with angles of 30, 60, and 90 degrees
OA is 4
Therefore, the answer is x + 4 is 12
and x is 8
As we have seen above
right triangles, created by dividing equilateral triangles and squares in half
are useful in Euclidean geometry
To summarize
we can think of two types of right triangles
a 30 degrees, 60 degrees, 90 degrees right triangle
and a 45 degrees, 45 degrees, 90 degrees right triangle
and the ratio of the lengths of the two right triangles is as follows
Let's solve the problem again
using these two right triangles
Question 4. Find the area of ​​an isosceles triangle
where AB is 8, AC is 8, and angle B is 75 degrees
If we look into the information about the given shape a little more
since angle B is 75 degrees, angle C is also 75 degrees
and angle A is 180 degrees - 75 degrees x 2
- 150 degrees, which is 30 degrees
Looking at the information written like this
we need to think about angle B as 75 degrees
as 60 degrees plus 15 degrees
Let's draw line segment BD
from point B to AC to find point D
where the angle ABD is 60 degrees
If we draw lines like this, we can see that triangle ABD
is a familiar 30 degrees, 60 degrees, and 90 degrees
right triangle
The ratio of the lengths of 30 degrees, 60 degrees, and 90 degrees right triangles
is AB : BD, which is 2:1
So BD is 4
and BD is the height of hypotenuse AC in triangle ABC
In conclusion, the area of ​​triangle ABC can be calculated as follows
1/2 x 4 x 8
which is 16
Let me emphasize again that the only way to figure out what you don't know
is to approach what you don't know through what you know
In Euclidean geometry, you don't need to know much
If you can utilize just the special right triangles 30 degrees, 60 degrees, and 90 degrees
and right triangles 45 degrees, 45 degrees, and 90 degrees
you can solve many problems
Finding the special right triangles we need is the key to solving the problem
Here's another question
Question 5. What is the area of ​​a right triangle with a hypotenuse of 2 and one angle of 15 degrees?
The problem is about a right triangle with an angle of 15 degrees
As I mentioned, we know right triangles with angles of 30 degrees, 60 degrees, and 90 degrees
but we are unfamiliar with 15 degrees
Let's think about how we can utilize the right triangles
with angles of 30 degrees, 60 degrees, and 90 degrees that we know
Let's think about a triangle whose vertex is an angle of 30 degrees
by copying the given triangle and pasting it to the side, as shown below
If we think about a triangle with one angle of 30 degrees like this
we can find a special right triangle with angles of
30 degrees, 60 degrees, and 90 degrees
The ratio of the hypotenuse length and the shorter side of this right triangle is 2:1
So the area of ​​triangle ACC' is
1/2 x 2x 1
which is 1
The area of ​​triangle ABC that we are looking for will be half of this
1/2 is the answer
Calculating and Imagining
When we look at a math problem
we first think about
how to calculate the x and y variables
However, before doing such calculations
we need to think and imagine something
When we think of math, we often think of complex calculations
but that is not true math
The process of thinking about various things and imagining new things is
the process of first going through that process
and then processing it, which is what calculations are needed for
I think this applies similarly to you, digital content creators
Rather than just sitting in front of a computer and starting work
it would be more effective first to imagine
what kind of work you will do and how you will do it
We are currently learning Euclidean geometry from ancient Greece
but the ancient Greeks did not have calculation skills
The decimal system that we use today is a very advanced form of positional description
We can easily do various calculations, not just the four basic operations
but the ancient Greeks found even multiplication and division very difficult
So, I think they spent more time thinking logically while imagining things rather than calculating
And such imagination and logic
became the cornerstones of today’s civilization
For example, let's think about this problem
What is the area of ​​the shaded area of ​​the square
with a diagonal length of 10 in the following figure?
If you ask middle and high school students about this problem, most of them will do the calculation
How did the ancient Greeks, who could not calculate, solve this problem?
So, I think they would have solved this problem
logically using their imagination
Let's think about copying the shape given in the question
and pasting it like this
If you paste it like this, you can see that the area we are looking for
is one-fourth of a square with a side length of 10
The area of ​​a square with a side length of 10 is 100
The ​​shaded area is 25
You should do the calculation when necessary
However, it is a good way to solve the problem
to imagine before calculating
and approaching it in various ways
It is not a good habit to just calculate
Here's a question
Question 6. Find the area of ​​the given shape
The given shape can be considered a square
and a 30 degrees, 60 degrees, and 90 degrees right triangle
Let's divide the 30 degrees, 60 degrees, and 90 degrees right triangles
in the problem into four smaller
30 degrees, 60 degrees, and 90 degrees right triangles
If the two sides of the 30 degrees, 60 degrees, and 90 degrees right triangles are AB
the hypotenuse will be 2a
The hypotenuse of the smaller 30 degrees, 60 degrees, and 90 degrees right triangle created in this way
is the same length as one side of the left square
Therefore, if you put them together
the entire thing becomes one square
The condition of the problem is that 2a + 2b is 6
The side length of a new square made by cutting and pasting
four small rectangles is a + b
which is 3
Therefore, the area of ​​the newly created square is
3 squared
9
The area of ​​the shape we are looking for is 9
Solving the same problem differently
is a great way to gain different perspectives
and think of new things
Let's solve this problem in another way
First, let's look at the following picture
If you look at a right isosceles triangle
and an isosceles triangle with an angle of 30 degrees
we can see that when the two shapes have the same base
their heights are also the same
If the bases of the two triangles are 2a
the heights of the two triangles are both a, which is the same
Since the bases and heights are the same, the areas of the two triangles will also be the same
Let's apply this fact to this problem
If you draw a line parallel to the line connecting AD
once through point A
and if you call the point where that line intersects the extension of CD, A'
triangle BA'D becomes an isosceles triangle with angle A'BD of 30 degrees
Since the areas of the right isosceles triangle ABD
and the isosceles triangle BA'D are equal
Question 6 can now be solved by finding the area
of ​​the larger isosceles triangle CBA'
In summary, this problem has changed to finding the area of ​​an isosceles triangle
with a side length of 6 and an angle of 30 degrees
We have already solved finding the area of ​​an isosceles triangle with an angle of 30 degrees
in Question 4
If we solve it using the same method, we can find
30 degrees, 60 degrees, and 90 degrees right triangles
We can see that it is an isosceles triangle with a base of 6 and a height of 3
Therefore, the area is 6 x 3 x 1/2
is 9
I will introduce one more problem that requires an idea and amazing insight
It is a problem that requires you to find a familiar shape
When you see the solution
you will exclaim
Wow, there's something like this too
Give it a try
Question 7. Find the area of ​​the shaded triangle in the following square
This problem makes me think that there is too little information given
It is a bit absurd
It is necessary to pay attention to the fact that a square and a right triangle are given
You can copy the right triangle of the square
and paste four of them inside
Here, we need to look at the two right triangles
related to the shaded part
The shaded triangle ADE can be understood as a triangle with a base AB of length 6
and a height of 6
Does it catch your eye?
I think it would be good to understand by looking at this picture
Therefore, the area is 6 x 6 x 1/2
the answer is 18
If you take a math test, you need to answer quickly, calculate accurately
and write the answer within the given time
However, it is not important to quickly come up with one answer when studying math
If you only come up with one answer quickly, there are many cases where you do not study much
If you think about one problem in various ways and solve the problem while thinking deeply
you will study well
In that sense, rather than simply calculating
thinking about various things and logically examining them
will be a real study
I think this attitude is necessary for those who create digital content
Rather than just making something that is set
the basic attitude of a creator is to think about something else
and approach it more diversely
If you think of it in terms of art and business
a digital content creator does art plus business
If producing results is business
then thinking about various things in various ways is art
I hope you become a competent digital content creator who catches two rabbits
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
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
Calculating and imagining Before calculating to solve a problem, a process of thinking and imagining is necessary
Mathematics is not about doing complex calculations but rather performing calculations in the process of processing various thoughts and new imaginations
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