What Is Pythagoras Theorem? | PYTHAGORAS THEOREM | The Dr Binocs Show | Peekaboo Kidz

Summary

This video script explores the Pythagorean theorem, a fundamental principle in geometry with real-world applications. It traces the theorem's history, noting its use in ancient civilizations like Babylon and India, predating Pythagoras. The script explains the theorem's statement—that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides—and illustrates it with a practical example involving a cat rescue scenario. It demonstrates how to apply the theorem to find the correct ladder length, emphasizing the theorem's relevance and utility in everyday situations.

Takeaways

• 📚 The Pythagorean theorem is named after Pythagoras of Samos, a Greek mathematician and philosopher, but he was not the actual founder of the theorem.
• 🌟 The theorem predates Pythagoras and was used by ancient Babylonians and Indians, as well as in Egyptian constructions.
• 📐 The 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.
• 🔢 The Pythagorean equation is expressed as \( a^2 + b^2 = c^2 \), where \( a \) and \( b \) are the legs of the triangle, and \( c \) is the hypotenuse.
• 📏 The theorem can be applied to real-world problems, such as determining the length of a ladder needed to reach a cat stuck on a wall.
• 🐱 An example in the script illustrates using the theorem to find the correct ladder size by measuring the lengths of the wall and the ground, and then calculating the hypotenuse.
• 🧮 To find the ladder's length, one must square the measurements of the wall and the ground, add them together, and then take the square root of the result to get the length of the hypotenuse in meters.
• 🛠️ The script demonstrates how to apply the theorem step by step, including the calculation of squares and the use of the square root to find the practical solution.
• 📈 The script also touches on the historical and cultural significance of the theorem, mentioning the Pythagorean society and their contributions to science and philosophy.
• 🌱 The Pythagoreans believed in the equal rights of animals, which led to their prohibition of eating animals, showing a connection between their mathematical pursuits and ethical beliefs.
• 🎓 The episode concludes with an invitation to learn more in future episodes, emphasizing the educational purpose of the content.

Q & A

Who is the Pythagorean Theorem named after?

The Pythagorean Theorem is named after the Greek mathematician and philosopher Pythagoras of Samos.

Did Pythagoras invent the Pythagorean Theorem?

No, Pythagoras did not invent the Pythagorean Theorem. He is credited with proving it, but the theorem existed even before him.

In which ancient civilizations was the Pythagorean Theorem used?

The Pythagorean Theorem was used in ancient Babylon, India, and Egypt.

What does the Pythagorean Theorem state?

The Pythagorean Theorem states that the square on the hypotenuse of a right-angled triangle is equal in area to the sum of the squares of the other two sides.

How is the Pythagorean Theorem expressed mathematically?

The Pythagorean Theorem is expressed as a^2 + b^2 = c^2, where a and b are the legs of the triangle and c is the hypotenuse.

What are the legs and hypotenuse of a right-angled triangle?

In a right-angled triangle, the legs are the two sides that form the right angle, and the hypotenuse is the side opposite the right angle and the longest side of the triangle.

How can the Pythagorean Theorem be applied in a real-life scenario?

The Pythagorean Theorem can be used to determine the length of a ladder needed to reach a certain height on a wall when the distance from the wall to the base of the ladder is known.

How do you calculate the hypotenuse using the Pythagorean Theorem?

To calculate the hypotenuse c, you take the square root of the sum of the squares of the other two sides, c = sqrt(a^2 + b^2).

What was the length of the wall and the land in the provided example?

In the provided example, the length of the wall was 3 meters, and the length of the land was 4 meters.

What is the significance of the Pythagorean Theorem in construction?

The Pythagorean Theorem is significant in construction as it helps in determining accurate measurements and ensuring structures are built correctly, such as in the construction of the pyramids in ancient Egypt.

What values did the example provide for a^2 and b^2?

In the example, a = 3 meters and b = 4 meters. Therefore, a^2 = 9 and b^2 = 16.

What is the final step to find the length of the hypotenuse in the example?

The final step is to take the square root of c^2, which in the example is sqrt(25) = 5 meters.

Outlines

00:00

🎉 Winning and the Introduction of the Pythagorean Theorem

The script opens with a celebratory exclamation of winning and a brief conversation about reading quickly, attributed humorously to the Pythagorean theorem. The narrator introduces the topic of the Pythagorean theorem and its applications in real life. The historical context is provided, noting that although the theorem is named after Pythagoras, it was known and used long before him, particularly in ancient Babylon, India, and Egypt. The theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

05:00

🐱 Saving a Cat with the Pythagorean Theorem

The narrator presents an example to explain the Pythagorean theorem. A cat is stuck on a wall, and the narrator needs to find the right-sized ladder to rescue it. The lengths of the wall and the ground are measured as 3 meters and 4 meters, respectively. Using the Pythagorean theorem formula \(a^2 + b^2 = c^2\), the narrator calculates that the ladder (hypotenuse) should be 5 meters. The process involves squaring the lengths, summing them, and then taking the square root of the result. The successful rescue of the cat with the correctly sized ladder demonstrates the theorem's practical application. The script concludes with a fun fact about the Pythagorean society's beliefs and compassion towards animals.

Keywords

💡Pythagoras Theorem

The Pythagoras Theorem is a fundamental principle in geometry that states the relationship between the sides of a right-angled triangle. It asserts that 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. In the video, this theorem is used to illustrate how mathematical formulas can be applied in real-world scenarios, such as determining the length of a ladder needed to save a cat stuck on a wall.

💡Hypotenuse

The hypotenuse is the longest side of a right-angled triangle, which is opposite the right angle. In the script, the hypotenuse is used as an example to demonstrate the application of the Pythagoras Theorem, with the ladder's length representing the hypotenuse in a practical situation.

💡Right-Angled Triangle

A right-angled triangle is a type of triangle that has one angle measuring exactly 90 degrees. The script uses the right-angled triangle to explain the Pythagoras Theorem, emphasizing its geometric properties and how they relate to the theorem's formula.

💡Pythagorean Equation

The Pythagorean Equation, a^2 + b^2 = c^2, is the mathematical representation of the Pythagoras Theorem. It is used in the script to show how to calculate the length of the hypotenuse when the lengths of the other two sides are known, as seen in the ladder example.

💡Legs of a Triangle

In a right-angled triangle, the legs are the two sides that form the right angle. The script refers to the legs as 'a' and 'b' in the Pythagorean Equation, and their lengths are crucial for determining the length of the hypotenuse.

💡Square

In the context of the video, 'square' refers to the mathematical operation of multiplying a number by itself, which is used to calculate the areas of the sides in a right-angled triangle according to the Pythagoras Theorem. The script illustrates this with the example of squaring the lengths of the legs to find the area of the hypotenuse.

💡Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. In the script, the square root is applied to the calculated area of the hypotenuse to find its actual length, which is essential for determining the size of the ladder needed.

💡Pythagorean Clan

The Pythagorean Clan refers to a group of philosophers and mathematicians who were followers of Pythagoras. The script mentions that while Pythagoras is often credited with the theorem, it was actually a collective effort of this clan, and the true founder remains unknown.

💡Ancient Babylonians

The Ancient Babylonians are highlighted in the script as early users of the Pythagoras Theorem, indicating that the theorem's principles were known and applied in construction thousands of years before Pythagoras.

💡Egyptian Constructions

The script mentions that the Pythagoras Theorem was used in Egyptian constructions, particularly in building pyramids, showcasing the theorem's historical significance and practical application in ancient architecture.

💡Pythagorean Society

The Pythagorean Society, as mentioned in the script, was a main scientific school in ancient Greece that evolved from the Pythagorean Clan. It is noted for its belief in the equality of animal and human life, reflecting the broader cultural and ethical implications of the Pythagorean tradition.

Highlights

Introduction to the Pythagorean theorem and its real-world applications.

Explanation of the historical origins of the Pythagorean theorem, including its use in ancient Babylon, India, and Egypt.

Clarification that Pythagoras did not invent the theorem but provided its proof.

Statement of the Pythagorean theorem: The square on the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides.

Introduction of the Pythagorean equation: a² + b² = c².

Visualization of a right-angled triangle with sides labeled as a, b, and hypotenuse c.

Example scenario: Using the Pythagorean theorem to find the correct ladder size to rescue a cat.

Measurement of the wall (3 meters) and the land surface (4 meters) as the sides a and b of the triangle.

Substitution of the measured values into the Pythagorean equation: 3² + 4² = c².

Calculation of the squares of the sides: 3² = 9 and 4² = 16.

Summing the squares of the sides: 9 + 16 = 25.

Conversion of c² into c by taking the square root: √25 = 5.

Finding the correct ladder size to be 5 meters.

Verification of the theorem with the calculated values.

Mention of the Pythagorean society's contribution to scientific progress in Greece and their belief in animal rights.

Transcripts

oh

[Music]

yay

i won i won kitty oh doctor how do you

read so fast

it's simple the pythagoras theorem

helped me to get the fastest way home

from the market

the pythagoras theorem kitty

hey friends we often hear people asking

if math formulas could be applied in the

real world

so in today's episode let us explore the

formula which can be used in many

circumstances and learn how to solve the

pythagoras theorem

zoom in

so what is the pythagoras theorem

let us begin from the beginning

the phytogas theorem is said to get its

name from a greek mathematician and

philosopher pythagoras of samos who is

believed to be a pioneer of this theorem

but that's not true as he is not the

actual founder of this theorem but was

one of the pythagorean clan members who

devoted their lives to the subject and

never revealed who its real founder was

however in reality this theorem existed

even before pythagoras and was useful

even then

whereas pythagoras only gave us the

proof of this existing right angled

triangle theory

yes the traces of pythagoras theorem

could be found in the construction of

ancient babylonians which existed even

thousands years before pythagoras

not only that but it marks its presence

in ancient india as well

which is believed to have a rich culture

of mathematics

the mathematical miracle was also used

in egyptian constructions where they

used pythagoras theorem to build the

pyramid and other structures

but the crucial question is what does

the pythagoras theorem actually say

the pythagoras theorem says that the

square on the hypotenuse of a right

angled triangle is equal in area to the

sum of the squares of the other two

sides

this theorem can be written as an

equation relating the length of the

sides a b and c

often called the pythagorean equation

that is denoted with the formula

a square plus b square is equal to c

square

confused no worries let us look at each

of its terms with the help of an example

but before we start with our example we

need to learn a few terms first which

are essential to understand

now visualize a right angled triangle

which has a b and c sides

in this diagram a and b are the legs of

this triangle and the side opposite to

the right angled triangle which is c is

called the hypotenuse

and is the longest side of the triangle

[Music]

moving further to our example

imagine a cat stuck on the surface of a

wall screaming for help

and the only way to save the cat is by

climbing over the wall

so you'll look around and find a few

ladders lying beside

but you have less time in hand and the

cat is hanging between life and fall

so we need to choose the correct sized

ladder as quickly as possible

fortunately you remember the pythagoras

theorem that says

a square plus b square equals c square

so let us assume the wall as a

the surface of land as b

and the ladder you need to find

as c

which is also your

hypotenuse then you quickly take out the

measurement tape and measure the length

of the wall that comes to three meters

then you measure the surface of the land

that comes down to four meters

now as a equals 3 meters

and b equals 4 meters

then we know c will be equal to the sum

of the square of the length of the other

two sides

and to prove that

you need to substitute these values into

the formula which is a square plus b

square equals c

which will also give you the correct

size of the ladder

hence proving this theorem

now as you have the value of a and b

you are required to find their square to

move further

so the first thing you do is substitute

a as 3

and b as 4

and the values you get after

substitution are

9 and 16 respectively

which are their squares

then you put these square values into

the formula which adds up to 25 square

meters

which is the value of c square

but let me tell you that this is not the

end because the ladder size could never

be in square meters as the other two

sides are in meters

so our next task is to convert this c

square into c

and its value square meter into meters

and to do so we apply square root on

both sides which converts c square into

c

and 25 square meters into 5 meters

and hence the value of the hypotenuse

and the correct size of the required

ladder

is

5 meters

and along with that two things happen

here

one is you prove the formula right as

that the square on the hypotenuse of a

right angled triangle which is c square

whose value is 25 square meters is equal

in area to the sum of the squares of the

other two sides

which are a square and b square

that is

9 plus 16

is also 25

and the second thing we found is the

correct size of the ladder which is five

meters

and hence you can save the cute little

kitty cat easily

trip your time

did you know the pythagorean society

later progressed as the main scientific

school of greece

also pythogrians think that animals have

the same rights to live as mankind hence

they held the prohibition of eating

animals

so kind of them

hope you learned something new in

today's episode until next time it's me

dr bynox zooming out

[Music]

oh

never mind