WEBVTT

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Euclidean Geometry

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Logic and Problem Solving

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Euclidean Logical Thinking for Digital Content Creators
7. Problem-Solving Techniques to Break Free from Conventional Answers Part 2

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Hello, this is Jongha Park

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In this session, we will approach problems from a new perspective

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introducing a problem-solving technique that involves actively

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transforming problems

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Everyone has their own framework of thought

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We rely on common sense and think within certain boundaries

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which helps us navigate everyday life efficiently

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However, as times change and circumstances evolve

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past conventions often fail to apply today

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What was right yesterday may be wrong today

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and what was wrong yesterday might be right today

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We live in a world where such shifts are increasingly common

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In such a world, flexibility of thought and freedom from stereotypes

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are essential

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For you, digital content creators

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it’s crucial to propose new thought frameworks

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and guide your content consumers

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This session will help you experience and learn techniques

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for breaking free from conventional thinking

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Approaching Problems from a New Perspective

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Let’s try a quiz

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In the matchstick equation given below

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move only one matchstick to make the equation correct

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There are three solutions

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Try finding all three

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This quiz is not related to math questions

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but it helps broaden your thinking and develop techniques

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 for solving math problems

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Approach it with flexible thinking

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and try to find all three solutions

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Pause the video, solve the problem

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and then resume to check your answers

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The answers are as follows

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Have you confirmed the solutions?

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Now, with this flexible thinking, let’s solve a Euclidean geometry problem

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Problem 1: Calculate the combined area of the two shaded triangles in the figure below

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This problem becomes straightforward once you understand the situation

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However, without flexible thinking, finding the answer can be very challenging

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Here’s how you can understand the problem

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The area of a triangle is base x height x 1/2

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Triangles with the same base and height

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have the same area, regardless of their shape

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Now, in the given figure, identify triangles

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with the same area as the shaded ones

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For example, the red triangle and the shaded triangle have the same area

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because they share the same base and height

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The two red triangles perfectly divide the rectangle into halves

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Thus, the combined area of the red triangles is

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5 x 8 x 1/2

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= 20

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This is also the combined area of the shaded triangles

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Solving one problem

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allows us to gain experience with related scenarios

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Just like gaining experience points in a game helps you level up

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these problem-solving experiences enhance your abilities

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Let’s use what we learned from Problem 1 to solve another problem

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Problem 2: In the figure below, there is a shaded triangle within the hexagon

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The total area of the hexagon is 24

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What is the area of the shaded triangle?

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Begin by examining what proportion of the hexagon

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the shaded triangle occupies

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The shaded part lies entirely in

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the upper half of the hexagon

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Let's look at the upper part first

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We are looking at the colored triangle

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But let’s shift focus to the unshaded triangle

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Instead of seeing it as

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a white background with a blue triangle

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consider it as a blue background with a white triangle

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If you look at the white triangle

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you can also think of it as the following triangle, as they share the same base and height

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By looking it like this, we can see that the white triangle

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is 1/3 of the upper half of the hexagon

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Therefore, the shaded triangle

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constitutes 2/3 of the upper half

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or 2/6 of the entire hexagon

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Given that the hexagon’s total area is 24, the shaded triangle’s area is 8

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When solving math problems, there is something called a "model answer"

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It refers to the method taught by teachers

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or presented in textbooks

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One thing to keep in mind is that the model answer is not the only correct answer in the world

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It's better to solve math problems using various methods

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The more ways you solve problems, the more your math skills improve

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So, instead of stopping at the model answer

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it’s good to solve problems in various ways, including the model answer

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If you're taking a math test, you'll need to find the answer quickly using a single method

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However, when studying math, try solving a single problem in multiple ways

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The more ways you attempt

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the more effective your learning will be

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Try the model solution, and also experiment with unconventional methods

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In particular, if you frequently use what may seem like shortcuts or tricks, you'll develop a sense of intuition

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Mathematical intuition is very important

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Let’s examine this concept through a problem

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Problem 3: In the given figure, calculate the area of the shaded triangle

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Some people say they solved this problem in 5 seconds

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They didn’t just calculate randomly

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they came up with an insightful idea

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Let’s solve this problem using two methods

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First, let’s approach it with the model solution

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This problem requires using the fact that the two right triangles, ABC and EDC, are similar

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By identifying corresponding sides, we can set up a proportion

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AB/BC = ED/DC

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From this, we can perform the following calculations

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Ultimately, BC × ED = 6 × 4

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The area of triangle BCE is half the product of the lengths of BC and ED

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Therefore, the area of triangle BCE is 1/2 x 6 x 4

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= 12

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But let’s think about this differently

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As explained in the previous session

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the areas of the shaded region (S) and S’ are equal

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Looking at the figure

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this is because the remaining parts of two triangles

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with equal areas are also equal

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With this thought, let’s reexamine the problem

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In the given problem, draw a line connecting AD

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to create triangle ACD

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Now compare triangles ABE and ABD

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Both triangles share the same base, AB

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and since AB is parallel to DE, their heights are the same

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Thus, the areas of triangles ABE and ABD are equal

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If you subtract the common area ABC from these two equal-area triangles

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you can conclude that the areas of triangles BCE and ACD are the same

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Triangle ACD has a base of 4 and a height of 6

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Thus, the area of triangle ACD is 1/2 x 4 x 6

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= 12

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which is the same as the area of triangle BCE

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Let’s continue solving problems

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Problem 4: Calculate the area of the shaded right quadrilateral in the given figure

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This problem can also be approached in two ways

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For a quick calculation

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use the similarity between triangles ADF and FEC

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From the similarity ratio, you can find the product of DF and FE

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Since the area of a right quadrilateral is simply the product of its base and height

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DF × FE gives the area of the quadrilateral

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The similarity ratio is calculated as follows

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The area of right quadrilateral DBEF is 36

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But let’s look at the problem differently

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The given right quadrilateral

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can be thought of as a part created by the diagonal of a rectangle

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Thinking this way

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A and B are equal, and C and D are also equal

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Thus, the areas of the two shaded regions are equal

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The area of the rectangle marked in red is 3 x 12

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which is 36

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so the area of the quadrilateral we’re looking for is also 36

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When solving Problems 3 and 4

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some people claim they can find the answers

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within 3 seconds

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If we solve the problems using what we call “shortcuts"

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it might indeed be possible to find the answers in such a short time

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While I’ve referred to these methods as the model answer and the shortcut

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thinking in terms of shortcuts is important as well

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It reflects a deep understanding of the problem

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and brings innovative ideas to problem-solving

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The key is to understand the problem from various perspectives

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and to develop a sense for solving problems

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One thing to be cautious about

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A “shortcut” should not mean a guess made

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without logical rigor

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or clear reasoning

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Even when using shortcuts

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you must maintain logical thoroughness and base your reasoning on solid grounds

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Shortcuts do not mean guessing blindly

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they mean finding a quicker path than usual

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through intuition and sense

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For digital content creators

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demonstrating a sense of intuition while working is even more critical

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Can there be a model answer when creating something new?

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True creation is implemented through

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the creator’s sense of intuition

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Remember the importance of intuition

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Let me introduce another problem

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Problem 5: Find the area of the square in the following diagram

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This problem requires setting variables and performing calculations

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You’ll need to apply concepts like the Pythagorean theorem from the textbook to solve it

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However, before diving into complex calculations

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let’s solve it using the following method

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Divide the given figure like this

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If divided this way, the entire right isosceles triangle is split

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into nine smaller right isosceles triangles that are identical in shape and size

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The square we want to find the area of consists of four out of these nine parts

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Thus, the area of the square is

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4/9 of the total area

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The area of the larger right isosceles triangle is

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10 x 10 x 1/2

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=50

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So 50 x 4/9

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200/9 is the area of the square

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Actively Engaging with Problems

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When solving problems, instead of passively waiting for ideas to come to you

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you should actively engage with the problem

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For instance, in Euclidean geometry

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you might extend lines, draw auxiliary lines

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or even cut and attach sections of a shape to other parts

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Such active efforts are essential for problem-solving

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Sometimes, merely imagining cutting out

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and attaching a certain part to a suitable location

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can lead to a solution

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When a problem is solved in this way, it often brings an exhilarating sense of achievement

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During practice, try drawing lines or cutting and attaching shapes on paper

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to solve a problem or two

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By cutting and attaching paper while solving problems

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you can expand your perspective

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and enhance your imagination

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Problem 6: In the following scenario, calculate the area of triangle ABC

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A methodical approach would be

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to use the similarity of triangles and calculate the similarity ratio

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But let’s consider it differently

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Let’s think about triangle CDE

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Triangle CDE has the same area as triangle ACD because

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they share the same base and height

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By drawing and observing triangle CDE

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we find that the area of triangle ABC

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is the same as the area of triangle BDE

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Since BE = 10 and CD = 4

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the area of triangle BDE is 10 x 4 x 1/2

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= 20

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Thus, the area of triangle ABC is also 20

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Problem 7: Calculate the area of quadrilateral ABCD as given in the following conditions

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Among the problem's conditions, note that

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AM = BM

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Looking at M as the center AMD+BMC=90 degrees

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Thus, by cutting triangle AMD

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and attaching it to make AM equal to BM

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the entire figure forms a new triangle as shown below

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When cut and reattached this way the area of the quadrilateral we’re solving for

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becomes the area of triangle DCD′

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the area of triangle DCD′ is 12 x 8 x 1/2

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= 48

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Let's continue

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Problem 8: In the following diagram, AB = AD

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D + B=180º

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AC = 10

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and ACB=15º

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Find the area of quadrilateral ABCD

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Given the condition AB = AD

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and D + B = 180º

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by cutting triangle ACD and attaching it so that points B and D align

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we find that segments BC and DC fall on the same straight line

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When represented visually, it appears as shown below

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Since AC has been cut and reattached, AC becomes AC'

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Because it is an isosceles triangle, angle AC'B=15º

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Now, our task is to calculate the area of the isosceles triangle ACC'

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To find the area of triangle ACC'

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consider the right triangle on a continuous line as follows

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Since AC = 10, AC'=10 and EC'=5

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The area of triangle ACC′ can be calculated as the following

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1/2 x 2C' x AC = 25

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= 25

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By cutting, reattaching, and approaching the problem

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in various ways

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the moment a problem is solved brings a significant sense of accomplishment

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This sense of accomplishment stems from actively engaging with the problem

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It’s a feeling similar to the joy experienced by creators

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when they make something new

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Rather than passively fitting into a predefined mold

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all of your digital content creators

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should actively creating a new framework

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We study math not just to learn concepts and knowledge

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but more importantly, to develop problem-solving skills

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Everyone learns through experience

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Sometimes people tease those

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inexperienced in relationships by saying

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I learned about love through books

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While reading and acquiring knowledge is important

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learning through experience is even more valuable

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In this session, we’ve tackled many problems

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Through these experiences

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apply the problem-solving techniques you’ve learned to your own challenges

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As digital content creators

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we’ve explored Euclidean geometry across seven sessions

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As emphasized at the beginning, the most important thing is fun

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Did you have fun?

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By experiencing cognitive enjoyment, we hope you’ve not only enjoyed math and logical thinking

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but also naturally improved your problem-solving skills

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Apply these thinking techniques to your creative activities

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and achieve even better results

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Always remember: fun and confidence are key

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We cheer for you as digital content creators

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Approaching problems from a new perspective
Exploring problems through many methods enhances skills so solving through diverse approaches is important

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Trying various methods can enhance your problem-solving skills
You must create intuition for solving problems

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Even if you use a shortcut, it should be backed by logical and solid reasoning

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Actively Engaging with Problems
Actively engaging with the problem is crucial, such as drawing extended or auxiliary lines, and cutting shapes

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The moment you solve a problem after approaching it in different ways, you feel a significant sense of achievement

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Rather than passively conforming to pre-existing frameworks, creating new frameworks actively leads to a deeper sense of fulfillment