Pythagoras (1) - Pengenalan Teorema Pythagoras, Pythagoras Theorem - Matematika SMP

Summary

This educational video introduces the Pythagorean theorem, exploring its history and various proofs. It demonstrates how to apply the theorem to solve problems involving right triangles, including practical scenarios like ladders against walls and ships' movements. The presenter encourages viewers to explore more proofs online and practice solving related problems for better understanding.

Takeaways

• 😀 The Pythagorean theorem is introduced, stating that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
• 🏛️ Pythagoras, the ancient Greek philosopher, is credited with the discovery of the theorem, although the concept predates him.
• 📚 There are numerous ways to prove the Pythagorean theorem, with one website reportedly listing a hundred different proofs.
• 🔍 To solve Pythagorean problems, it's important to identify the right triangle and its angles, and then apply the theorem accordingly.
• 📐 The script provides examples of solving for missing sides of a right triangle using the theorem, emphasizing the importance of recognizing the hypotenuse and the other sides.
• 📈 The transcript illustrates the process of solving Pythagorean problems step by step, including how to handle calculations and what to look for in the problem setup.
• 🌐 It mentions practical applications of the Pythagorean theorem, such as calculating the height of a ladder against a wall or the distance a ship has traveled.
• 🚢 A real-world problem involving a ship sailing east and then north is used to demonstrate the application of the theorem in calculating distances.
• 👨‍🏫 The script encourages viewers to practice solving Pythagorean problems and to explore the many available proofs to better understand and apply the theorem.
• 📺 The video is designed to be educational and engaging, with a call to action for viewers to like, subscribe, and share the content.

Q & A

What is the Pythagorean theorem?

The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Who was Pythagoras and what is he known for?

Pythagoras was an ancient Greek philosopher and the founder of the Pythagorean Brotherhood, known for the Pythagorean theorem which is a fundamental principle in Euclidean geometry.

How is the Pythagorean theorem applied in the context of the video?

In the video, the Pythagorean theorem is applied to solve various mathematical problems involving right triangles by calculating the length of one side when the lengths of the other two sides are known.

What is the formula represented by 'bc² = ab² + AC²' in the context of the video?

This formula represents the Pythagorean theorem where 'bc' is the hypotenuse, and 'ab' and 'AC' are the other two sides of a right-angled triangle.

Can you provide an example of how to solve a Pythagorean problem as described in the video?

Sure, if you have a right triangle with sides of lengths 3 and 4, and you want to find the hypotenuse, you would calculate it as √(3² + 4²) = √(9 + 16) = √25, which equals 5.

What does the video suggest as an alternative to memorizing the Pythagorean theorem formula?

The video suggests visualizing the triangle and its sides to understand the relationship between them, which can help in remembering and applying the theorem.

How many different proofs of the Pythagorean theorem are there according to the video?

The video mentions a website that displays a hundred different proofs of the Pythagorean theorem, indicating there are numerous ways to prove it.

What is the significance of identifying the 'elbows' in a triangle when applying the Pythagorean theorem?

Identifying the 'elbows' or right angles in a triangle helps to determine which sides are perpendicular and thus which sides can be used in the Pythagorean theorem to find the hypotenuse or one of the other sides.

How does the video approach solving story problems involving the Pythagorean theorem?

The video suggests drawing the scenario to visualize the problem, identifying the right triangle, and then applying the Pythagorean theorem to find the unknown side.

Can you provide a real-world example given in the video where the Pythagorean theorem is applied?

Yes, one example is a ladder leaning against a wall where the ladder's length, the distance from the ladder's base to the wall, and the height up the wall where the ladder touches are used to form a right triangle. The Pythagorean theorem is then used to calculate the height up the wall.

What is the final advice given in the video regarding solving Pythagorean problems?

The video advises starting with the largest triangle when there are multiple triangles involved and to remember that the process involves looking at the 'elbows' or right angles to apply the theorem correctly.

Outlines

00:00

📐 Introduction to Pythagorean Theorem

The script introduces viewers to the Pythagorean theorem through the Le Gurules song channel. It explains that the theorem was named after Pythagoras, who is depicted as an old man often lost in thought. The theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. The script suggests that there are many ways to prove this theorem, and viewers are encouraged to explore these proofs online. The presenter also discusses how to solve Pythagorean problems by identifying the right angles and applying the theorem formula to find unknown sides of a triangle.

05:11

🔍 Solving Pythagorean Problems

This section of the script delves into solving Pythagorean problems by applying the theorem to find missing sides of triangles. It illustrates the process of identifying the hypotenuse and using the formula a² + b² = c² to calculate unknowns. The script provides examples where the values of the sides are substituted into the formula to solve for the missing side. It also touches on the importance of recognizing right angles and using the theorem correctly. The presenter simplifies the process by avoiding memorization of formulas and instead encourages a visual approach to problem-solving.

10:12

🚢 Applying Pythagoras in Practical Scenarios

The script moves on to apply the Pythagorean theorem in real-world scenarios, such as calculating distances and heights. It discusses how to use the theorem to find the height a ladder reaches on a wall, the viewing distance from a person's eyes to the top of a tower, and the shortest distance a ship has traveled from its starting point after changing directions. The presenter emphasizes the importance of drawing diagrams to visualize the problem and then applying the theorem to find the solution. The examples provided are meant to help viewers understand how the theorem can be used in practical, everyday situations.

15:15

🌐 Advanced Pythagorean Applications

In this part of the script, the presenter explores more complex applications of the Pythagorean theorem, such as calculating the shortest distance a ship has traveled after changing its direction multiple times. The calculations involve determining the ship's path by considering its speed and time traveled in different directions. The script demonstrates how to use the theorem to find the straight-line distance from the starting point to the final position of the ship, even when the path is not a straight line. This section aims to show viewers how the theorem can be applied to more advanced problems involving motion and distance.

20:16

🎓 Conclusion and Encouragement

The script concludes by summarizing the video's content and encouraging viewers to apply what they've learned. It suggests that the Pythagorean theorem can be used to solve a variety of problems related to stairs, towers, kites, airplanes, and ships. The presenter thanks viewers for watching and encourages them to share the video with friends, like the video, and subscribe to the channel for more educational content. The script ends on a positive note, emphasizing the value of learning and sharing knowledge.

Keywords

💡Pythagorean theorem

The Pythagorean theorem is a fundamental principle in geometry that states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem is central to the video's theme as it is the main mathematical concept being discussed and demonstrated through various examples and proofs.

💡Hypotenuse

The hypotenuse is the longest side of a right-angled triangle, which is opposite the right angle. In the context of the video, the hypotenuse is often referred to as the 'slanted side' and is crucial in applying the Pythagorean theorem, as it is the side that, when squared, equals the sum of the squares of the other two sides.

💡Right triangle

A right triangle is a triangle that has one angle that is exactly 90 degrees. This type of triangle is essential to the video's narrative because the Pythagorean theorem only applies to right triangles. The script uses right triangles in its examples to demonstrate how to apply the theorem.

💡Proof

A proof in mathematics is a logical argument for the truth of a proposition. In the video, various proofs of the Pythagorean theorem are mentioned, showcasing different methods to demonstrate why the theorem holds true. This keyword is important as it highlights the video's educational purpose of not just stating the theorem but also proving its validity.

💡Pythagorean problems

Pythagorean problems refer to mathematical problems that involve applying the Pythagorean theorem to find unknown sides of right triangles. The video script provides several examples of such problems, illustrating how to solve them by identifying the right triangle and applying the theorem.

💡Elbow

In the script, 'elbow' is used metaphorically to refer to the right angle in a right triangle. This term is used to help viewers identify which side is the hypotenuse when setting up a Pythagorean theorem problem, as the hypotenuse is opposite the right angle.

💡Story problems

Story problems are real-world scenarios presented as mathematical problems to solve. The video includes story problems related to the Pythagorean theorem, such as a ladder leaning against a wall and a ship sailing in different directions. These problems are used to demonstrate the practical application of the theorem.

💡Distance

Distance is a measure of the length between two points. In the video, distance is used in the context of calculating how far apart two objects are, such as the distance from a ship's starting point after it changes direction. This concept is integral to the story problems presented.

💡Speed

Speed is a measure of how fast an object is moving, typically measured in distance per unit of time. The video uses speed in the context of the ship's movement to calculate the distance it travels in a given time, which is then used in a Pythagorean problem to find the shortest distance from the starting point.

💡Factor

In mathematics, to factor means to express a number or polynomial as the product of its factors. The script mentions factoring in the context of simplifying the square root of a number when calculating distances, which is part of solving a Pythagorean problem involving a ship's journey.

💡Root

The root of a number is a value that, when an operation (like multiplication or exponentiation) is applied to it, gives the original number. In the video, the square root is used to find the length of the sides in right triangles after applying the Pythagorean theorem, as seen in the examples involving ladders and ships.

Highlights

Introduction to the Pythagorean theorem and stories about Pythagoras.

Description of Pythagoras as an old man who often thought and was a bit sad.

Explanation of the Pythagorean theorem: the square of the hypotenuse is equal to the sum of the squares of the other two sides.

The theorem only applies to right triangles.

Many ways to prove the Pythagorean theorem are available.

Simplest way to visualize the Pythagorean theorem using geometric shapes.

Encouragement to explore various proofs available on the internet.

How to solve Pythagorean problems with examples.

Importance of identifying the right angle in a triangle to apply the theorem.

Step-by-step approach to solving Pythagorean problems using the theorem.

Using the Pythagorean theorem to find missing sides of a triangle.

Explanation of how to handle problems with multiple triangles and right angles.

Solving problems by considering the largest triangle first.

Application of the Pythagorean theorem to real-world scenarios like ladders against walls.

Calculating height using the theorem with examples of ladders and walls.

Using the theorem to determine viewing distances, such as from a person to the top of a tower.

Solving motion problems involving ships sailing in different directions and calculating distances.

Encouragement to share the video and subscribe for more educational content.

Transcripts

Hello friends, see you again with the Le Gurules song channel. In this video, we will discuss

a new lesson, namely about the Pythagorean theorem. In this video, we will learn an introduction

to the Pythagorean theorem and also about stories about Pythagoras. So, Pythagoras, as his name suggests, was

discovered by someone named Pythagoras around in the first century, that is, he looked like

Pythagoras was described as an old man who often thought and/or was a bit sad,

right? Well, Pythagoras stated that the square of the slanted side of a triangle is equal to the sum of the squares of the upright sides

. So, the slanted side yes bc² = ab² + AC squared and only applies to right triangles

There are many ways to prove the Pythagorean theorem.

So, here's one of the ways we can see here.

There's the simplest way on the left and there's also a way to move the triangle

in the middle as well as move the triangle

.

So maybe if friends get an assignment to look for Pythagorean proofs, you can see a lot of them on the internet. There's

even one website. You forgot the name. It displays a hundred Pythagorean proofs

. How to solve Pythagorean problems, let's just try it right

away . triangular fruit, everything must have a pair of elbows, if there are no elbows, we can't assume it's a right angle, even though it looks as if we see that there are right angles.

Now we will complete

the side that doesn't exist yet, here we have BC, so it's empty, there is ml. blank here there is rq0 So actually

compared to memorizing the KK formula I prefer to see it like this why Because if it's Speed ​​3

It's going to be a headache going back and forth, where's the slanted side So the first time you see is whose elbows

are there in front Oh, there's BC in front so this one bc² = we just write down

the rest we are looking for because here it doesn't mean 4 squared plus 3 strike out so this is 16 added sum 9

bc² = 25 then you can get the pc is the root of 25 we try for the second question it's the same

so the first step to look at is the side of the elbow in front who is there Oh there is MK

means MK squared = lm² plus KL is left Let's just enter the number in the

square, how much is it, oh, 13 squared, right? Oh

, the rm doesn't have lm² plus 12 squared like that. This is 169 = LM squared plus 144,

you'll get the LM squared = 16944, the result is 25

. Oh

, there's PQ, so pq² = pr² + QR, we just need to enter pq², how much, oh, 25 squared

is the same as PR, PR is 15² + QR, we calculate this 625 = 225 + QR, then QR squared is 625

minus 225, we get QR squared is 400, so This QR is the root of 400 So

the result is 20, it makes a pretty good number so we can easily

solve it . Let's try the next problem. So you have a

slightly different triangle.

if we have a triangle like this then we can call the front side of angle a small a like that

in front of small BB maybe sometimes your teacher gives you the root formula uh c² = a² + b

squared for example like that Well, that's a problem, but if you prefer to look at whose elbows.

Well, here you have the sides now. Now, we want to make the first

one.

yes, that means you can't use Pythagoras d² = x² + a squared. We can't enter the steps

for the a, there are three si c, there's 8, then there's d, which is 10, so we want to find B and E if

we want to find e we are still in drawer E, there is a triangle where there is a triangle on the left,

there is also a straight line in the triangle on the right, but we will see what is there, what are the names of the elbows

, we refer to the e, so from here we will get x² = b² + c² use this triangle

, yes, this is very important, but our problem is that we only know the c

8² Sis, you don't know the e and now or the b.

brother put it here in the triangle the greater one

Okay, brother, look at the angles, but brother, shoot your eyes to the end. Oh

, it means that if you use a big triangle, D squared = c², yes, the angles c are equal to one more, a

plus b squared, right? Well, the square is 10 squared, C squared is 8² + a + b

squared then let it be like this here's 100 = 64 + a + b² yes 36 = a + b² I got

ya A plus B the squared is the root of 36 or 96 a + b Yes, but what we need

here is just the b is all we need, just B and AB, but we have a for

3, which means we can read this again 3 plus B = 6, so we get the b, which is 3

, right , 6 - 3 means x² = 3² + 8² 3 what color is 9 plus 8 squared 64 means the X squared

is 73 e is the root of 73 so let's just leave it like this now let's try again for

the b now we have only 9 we have b 6 we have e it's equal to 10, yes, that

's c and d, it's easy for D, we need a triangle not big. So to calculate

D, we make a big triangle. We calculate this one from this small triangle. Sis,

use the front angles, there are 10 or e, which means e² = b² + c², this is 10 squared = B squared, that

's 6 squared plus C squared, now 100 = 36 + c² 64 we get C for 88 Now we want to do

the math, so we use the big triangle, we use the big triangle,

bro, keep looking at the elbows, sis, look at the front.

the square plus the side of this isn't it just BA + b² Yes 8² + A + B means 15 yes this

is added is the result 15 squared gets 64 + 225 the squared is the same as

what is 289 yes this is 289 provided yes Well 289 you guys it turns out that if you root

it the result is good here's 17 cm, so this means 17 cm, okay, if it's a triangle, it's a

quadrilateral, right? Calculate the value of BC. If you find a question like this,

we can actually change it, force the questions to change a little, the method is bro, make g aris, now

white,

we have a triangle, now there are two triangles, one at the bottom and one at the top, OK

? Now we want to calculate the BC. Okay, let's see that BC has a triangle

. dc² + bc² yes DC

squared is 5 squared plus Wow BC is what you are looking for, OK, I don't know the difference,

sis. Okay, now we are looking for BD with another triangle, remember we still have this capital.

Sis, use the model on this side, now look at the front angles. it's the same

, bd² = perpendicular side means ad squared plus bd² uh AB squared, that's OK, so ad

squared is 6 squared plus 8 squared 36 plus 64 so you get 100

[Music] that's different means 100 = 25 + bc² we get the bc² is 75, so we get that

the bc is the root of 75, so we can

simplify the roots , OK? 15 Okay, how

much more is this 5 * 3, it means that BC is the root of 5² * 3 which

can come out to get 5 roots of 3 bc.

Here's what was asked AF again, the rear one doesn't

matter, we'll have to start from the biggest, right? We'll start with the biggest triangle,

we'll see the Def triangle, we'll look at the angles at the ends, this means EF

squared = what is De squared plus DF per F what is the square of it? 20 squared = 6² + df² means

this

is 400 = 6 is 36 + DF the square is If we move here, it means 364

. look at the opposite angles

plus CF how many 364

is 5 squared plus cf4 means 364 = 25 squared plus cf4, that's right Eh 25

if you move sides the result is 339 = cf² Just leave the cf d in that shape Oh why is it

because the type is using it like that while you are triangular bcf so here you have the B brother you

look at the end like that yes it's always like that so we have cf² = bc² + bf² The CF

is this 339 Sis don't have to count again = bc² its 4 squared plus our BF squared

means 339 = 16 + BF get the bf, that's the beve², it's 323

, the last thing we just counted is AF, let's look at the angles in A,

the front is BF squared = ab² + af², BF squared is 323 = ab² 3² + HF = 9 +

that is 323 - 9 the result is 314

is the root of 314 Let's just leave it like this if you tell it it's okay to end up like that

if you meet a problem like this don't be confused okay Usually we start counting the one that

is the last one from the biggest Bro Then the DF is squared don't count

it so we can use it right away, if you guys are that later, you've already mastered it

again, it's a waste of time, it turns out that the results are still the same, OK? Now we want to try

some story problems from Pythagoras.

Sis, draw so we know what the triangle looks like

. If we don't draw, we won't be able to draw it

. normal run, so a ladder with a length of 2.6 meters is leaning against a wall. If the

distance from the bottom end of the ladder to the wall is 1 meter, then how high is the ladder measured in the middle? That

means you made the wall first. Siblings made the wall. Brother owns the land,

now there are stairs there. here the figure is like this, the length is 2.6 meters, while from

this foot from the end of the stairs to the wall it is 1 meter. Now, for the wall, it is impossible for people to make

this wall, but we assume it is a right angle, so the height of the wall is the height

of the wall, actually what is being asked So, how high is the wall, beautiful? Let's see

from the angles. Let's see who's in front. Oh, 2.6 meters means 2.6 squared, right on the

left and right means 1² + how much is 2.6 squared, how much is it, let's calculate the result If 26

is 676, it means if there are 2 commas, this is how you do it like this Sis,

you just make 26 squared, how much is 6 7 6, but because the comma is a there's a comma,

it's in the quadrant too, so it's shifted by 2 like that, so this is the idea 6.76 = 1 + t² this means 5.76 =

the root of 5.76 so every day the results are good 2.4 m so the height is 2.4 m So, the length of the ladder is 2.6 meters, the distance between the feet is 1 meter, so

if we draw it, we know

what the situation is and we can solve the problem. Another example, the next problem is that a

soldier has a height of 46 49.6 stands looking at the top of the tower at a distance of

14 meters. The height of the palm fiber is 1, 6 If the child's line of sight is what is the child's line of sight or the

palm fiber to the top of the tower, how about now we have a tower, we draw the tower first. Brother, I don't

really like to draw well, so sorry if the tower is a bit weird, right? Brother, I have a BTS tower

, now the palm fiber is here

, look again up, we don't draw the line of sight, the point of view

is that if we look up, there's a laser from our eyes

, where do we think it's going to go here

, the tower is high, the tower is tall So he said that the height from here to

here is 49.6 meters, but this palm fiber is not very tall, he is only 1.6 meters

away. From here to here it is 14 meters. So, what are you asking? What is

the viewing distance? what is the viewing

distance that is being asked, how much is this, what

is the distance to here ?

perpendicular to the ground and then this, right? So we have here, it's still 14 meters

, so this is part 9.6 minus 1.6, how much does it get 48, 48 meters,

bro? this means visibility, for example

, means visibility squared = 14² + 48 squared = 14 that's 196 plus 48. This 48 squared is

JP squared, that's 2,500.

Ara is 49.6 meters tall, the palm fiber sees from 14

meters her line of sight is 50 meters. Well, let's try one

more example question, so there were two story questions about stairs and about tower visibility, you can also

later kites, airplanes and so on, the problem can be varied, but roughly the method is like that.

Now, let's try about the ship here. Sis, you have a ship, the Hati Ijup ship and the name of the ship

is Liver Live, sailing eastward at a speed of 80 km/hour for one and a half hours later the ship

rotates towards the north with a speed of 70 km/hour for 1 hour 24 minutes what is the

shortest distance now the ship from the beginning now let's draw it first, East means to the

right, west northwest, ok East to the right, then he walk to the right, for example

, the initial position of the ship is here, he walks here at 80 km per hour for one and a half hours

, now this is actually more a matter of motion, yes, maybe it was moved

. times t means 80 times one and a half gets

120 km we first write here 120 km then he moves up now he now moves

up he turns like that ok he turns up turns up the speed decreases to 70 km per hour

here's the v but the time is one hour 24 minutes yes this one means 24/60 2/5 hours yes too we

can calculate the s of V times the t the v is 70 times 1 2/5, that means 70 multiplied by 7/5 so

the result is 98 KM. Now we will just calculate the shortest

distance .

and North for sure, this means we look at the angles, we

look at the front, it means that the distance is the same as the squared distance, so yes, the squared distance

is 120² + 98 squared, 120 squared means 14400 plus 98 squared is 96

04, if we add up the result, we add

14400 400

142424004 yes So the distance is the root of 24004 now this number is a

bit complicated so yes if we factor it it's a bit difficult it only gets 4 times 6001

this too k it's hard to divide by two it can't be divided by 3 it can't be 9 it can't be 7 it can't

be if it's stuck at the root Sis right now you can only get up to 2

roots of 6001 KM right Now it's something like that So the story questions are usually related to

stairs or towers or about ships sailing East to north to west to south

and so on like that Thank you friends for watching this video

If friends find the video useful Come on, share it with other friends so

you can learn together, of course the results will be better don't forget to like this video subscribe

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