Euclidean Geometry
Logic and Problem-Solving
Logical Thinking in Euclidean Geometry for Digital Content Creators 6 Breaking the Mold of the Right Answer - Problem Solving Techniques 1
Hello, this is Park Jongha
In this lesson, I will introduce problem-solving techniques
such as collecting information and imagining in various ways
In this situation, it is difficult and often useless
to talk about
applying these problem-solving techniques in a short story
The problems we face are
unlike the math problems we are learning now, so they are not immediately apparent
However, I think it is very important to think about the need for these things in everyday life
In that sense, it would be good to remember that collecting sufficient information in this lesson is necessary
You should think about various cases
in various ways
In particular, in the digital content market
where new technologies and services are created daily
you should always be careful when collecting new information
Transforming Given Information into Needed Information
When solving a problem, collecting all the necessary information is important
The more information you have
the easier and faster it is to approach problem-solving
If you lack information, you cannot solve the problem
no matter how many ideas you try
Therefore, it is effective to approach the problem by indicating information
that will likely be used to solve the problem
For example, let's look at this question
In the following figure, what is x's angle
when RT and TU are equal?
Under the condition of the problem that RT is TU
triangle RTU is an isosceles triangle
Since 180 degrees - 106 degrees is 74 degrees
the sizes of the remaining two angles of an isosceles triangle are 37 degrees each
Let's write this down in the problem
If you write down the angles in the problem like this
By the exterior angle theorem
angle x in triangle SVU is 32 degrees plus 37 degrees
equals 69 degrees
Math problems are not solved by eye but by hand
The first thing you should do by moving your hand
is to indicate the necessary information
As you display the information necessary for problem-solving in this way
you will often have the experience of seeing the information you have displayed with your eyes
moving your head
and coming up with ideas
necessary for problem-solving
I recommend that those of you who deal with digital content
also hold a ballpoint pen in your hand on a practice sheet
like an analog and express your thoughts in various ways
Writing something on paper with your hand often gives you ideas
so I recommend trying it at least once
Let's solve one more math problem
Question 1. Find the area of ​​the triangle given below
If we fill in a little more information that we know, it is as follows
First, we can see that the given triangle is isosceles
because the sum of the internal angles of the triangle is 180 degrees
Therefore, the length of two sides of the triangle is 5
Also, if we think of a right triangle with lengths of 3:4:5
if the base is 5
we can see that the triangle's height is 3
Therefore, the area of ​​the triangle is 5 x 3 x 1/2
which is 15/2
There is a saying, "There's more than one way to skin a cat"
You may think that you just have to solve the problem somehow
Rather than solving a complicated and difficult problem
it is better to solve the problem easily and simply
That way, you can reduce calculation errors and solve the problem quickly
To proceed with the problem-solving process easily and simply
you must use the information effectively
To do that, it is important to understand the necessary information
Let's take a look at a problem to figure out the necessary information
Question 2. Calculate the area of ​​the shaded right triangle given the following
Students who learned the Pythagorean theorem in school
will probably start with complicated calculations
such as calculating the length of the hypotenuse of a large right triangle
However, I told you that blindly calculating
does not lead to good results
Before calculating, let's take a look at the length ratio
The length ratio of the simplest right triangle made of natural numbers is
3:4:5
Therefore, all triangles with a 3:4:5 ratio
are right triangles
The given large right triangle also has two sides of 9 and 12
This means that 9:12 is 3:4
Therefore, even without applying the Pythagorean theorem to calculate the hypotenuse
we can see that it is 9:12:15
If we solve the problem
first, we can see that the length of the hypotenuse of the large right triangle is
9:12:15
which is 15
If we write this
we can see that the length of the hypotenuse of the small right triangle is 10
If the lengths of the two sides of the small right triangle are a and b
then the right triangle is also a right triangle with a ratio of 3:4:5
so a:b:10 is 6:8:10
Therefore, the area of ​​the small right triangle can be calculated like this
1/2 x a x b is
1/2 x 6x 8
which is 24
Question 3. Find the size of the given angle x in a regular octagon as follows
Let's approach this problem in two ways
The first way is to figure out the given angles without thinking about the idea
First, let's think about the size of the interior angles of a regular octagon
We can think of six triangles
inside a regular octagon as follows
Therefore, the sum of the interior angles of a regular octagon is 180 degrees x 6
which is 1080
Since the sizes of the eight interior angles of a regular octagon are the same
if we display the size of the angles in the given problem
we can display all the values ​​of the angles
If four angles make a square
45 + 225 + 45 + x is 360
Therefore, x is 45 degrees
Let's approach this problem with an idea
Let's think about the following triangle
If we think about a triangle that contains angle x like this
we can see that the two remaining angles
except x, are angles that divide the interior angles of a regular octagon in half
Because just as we can think of the center of a circle
we can think of the center of a regular octagon as a picture
So in the triangle we drew, the other two angles, excluding x
are angles that divide the 135 degrees angle of a regular octagon in half
Therefore, the sum of the two angles is 135 degrees
and x is 180 degrees minus 135 degrees
which is the answer of 45 degrees
Thinking Beyond the Obvious
One of the charms of Euclidean geometry is that
you can solve problems by imagining things that are not directly revealed in the problem
The problems given to us are
often part of the overall situation
So, just by drawing extension lines or auxiliary lines
and expanding the situation a little
you can often get ideas for solving the problem
Drawing lines that do not exist
sometimes copying the shape of the problem
and moving it to the side
is an active and positive approach that often gives you ideas for solving the problem
Let's look at the related content while solving the problem
Question 4. Find the area of ​​the pentagon ABCDE given below
The sum of the interior angles of a pentagon is 540 degrees
This is 180 degrees x 3
Therefore, if we find the value of angle E
it is 540 - (90 x 3 + 135)
which is 135 degrees
Also, 180-135 equals 45 degrees
If we understand this relationship
we can think of two right isosceles triangles
Since the two sides of an isosceles right triangle are the same length
we can determine the following information
The large triangle FGD is also an isosceles right triangle
Since the side length is 14
the area of ​​the large isosceles right triangle
is 14 x 14x 1/2
which is 98
and the areas of the two smaller isosceles triangles are
4 x 4 x 1/2
which is 8
6 x 6x 1/2
is 18
Therefore, the answer to the area of ​​the pentagon
is 98 - (8 + 18)
and the answer is 72
One thing we need to be careful about is that
we should not use our imagination
and do illogical calculations by eye
For example, some people solve this problem
by drawing a line connecting B and E
and thinking that BCDE is a right triangle
If you set it up like this, it is not logical
This means ABE is an isosceles right triangle
AB = AE = 6
and BE = 10
This is not logical if you think about the ratio of the lengths of the sides
of an isosceles right triangle
Please remember that you cannot solve math problems
with your logic
Question 5. What is the size of each x in the following figure?
So that the rectangle BCDE becomes a square
Let's take point E
Then, since angle ADC is 150 degrees
we can see that angle ADE in triangle ADE is 60 degrees
and that the lengths of sides AD and DE are the same
Therefore, triangle ADE is an equilateral triangle
with all angles at 60 degrees
Since triangle AED is an equilateral triangle, AE is 2B
and triangle AEB is an isosceles triangle
Therefore, angle BAE is angle ABE
and is 15 degrees
We are looking for the size of angle x
to be 60 - 15
which is 45 degrees
If we express it as calculating and imagining
we should be good at calculating and imagining
If we only focus on calculating quickly
and think of the process of imagining leisurely as a waste of time
it will be difficult to learn truly fun math
Our math skills will not improve much either
Our daily lives are the same
It is necessary to do things quickly and efficiently
but it is also necessary to think bigger and imagine new things by looking around
If you are working hard but are not getting much better than average results
you need to take your time and imagine more
rather than blindly making plans to do things faster
Question 6. Find the area of ​​the given rectangle ABCD
Let's solve this problem
How would you solve it?
It seems like we could solve this problem by quickly calculating
However, since we haven't learned much about calculation methods in this course
let's solve this problem by imagining
Let's copy the given shape and paste it
If we paste four of the same shapes like this
we can see that it becomes a square with a side length of 5
Therefore, the square area is 25
and since we pasted four shapes
the area of ​​one shape we are looking for is 4/25
Some people like to paste, and some people like to divide
If you like to divide
draw a line segment BD
Then, cut triangle ABD
and paste it so that AD becomes CD
Since angle A plus angle C is 180 degrees
DBB' becomes a triangle
and B' becomes an isosceles right triangle, which is B'D
All we need now is to find the area
of ​​an isosceles right triangle whose hypotenuse is of length 5
The area of ​​an isosceles right triangle
can be calculated by attaching four of the same ones
If we attach the four isosceles right triangles
we will create a square with a side length of 5
Therefore, the area of ​​the square is 25
and the area of ​​one isosceles right triangle that we are looking for
is 25/4
In other words, the area of ​​the shape we want to calculate is 25/4
We see that difficult problems can be solved
by simply examining them logically one by one while using various imaginations
without doing difficult calculations
This is how we experience the fun of math
Question 7. In the following figure, find the area of ​​the shaded area when AB is 6
Let's approach this problem not by calculating but by imagining
Let's think of the given shapes as tiles on the floor
If we think of them as tiles on the floor
we can attach them, as you can see on the screen now
If we attach four shapes like this
we can see a square with a side length of 6 is created
The area of ​​the given shape is 6 x 6
A quarter of a square with a side length of 6
so the answer is 9
his problem can also be solved using another analytical method
Since solving one problem in various ways is a good way to study
let's solve it using another method
In this problem, the area of ​​the shaded area
can be thought of as a square and a right triangle
It can be interpreted as the area of ​​3/2 + 1
5/2 squares
Therefore, if you find the area of ​​one square with the given condition
you can find the area of ​​the shaded area
by finding 5/2 of it
The area of ​​one square can be found as follows
The area of ​​the shaded triangle is 3/2
If you attach the shape on the left as on the right
you will create a large square with a side length of 6
which has 4 shaded triangles and 4 squares
Therefore, if we call the area of ​​one square A
we can set up the following equation
If we follow the equation you just saw, A will be 18/5
We said that the area of ​​the shaded area
is 5/2 times A
Since the area of ​​a square of the value of A is 18/5
the answer to 5/2a is 5/2 x 18/5
which is 9
I solved the problem by imagining rather than calculating
After learning the Pythagorean theorem
and solving quadratic equations in middle school
many problems are approached by calculating
Calculating this way can solve problems more easily
but sometimes, you will experience such calculations blocking your imagination
Since this course aims to solve all problems
with 4th-grade knowledge
I solved the problem by imagining rather than calculating
I hope this can provide insight to you, the digital content creators
Rather than quickly coming up with an answer
you can think of something else with various imaginations
Converting given information into necessary information The more information you have, the easier and faster you can approach problem-solving
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
Imagine a wider part The problems given to us are often part of the overall situation
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
Even without making difficult calculations, we can solve difficult problems by exercising our imagination and logically considering them