Teorema Pythagoras Kelas 8 Semester 2

Summary

This educational video script discusses the Pythagorean theorem for 8th-grade students. It explains the theorem's application in determining the type of triangle (acute, right, or obtuse) using side lengths. The script also covers the proof of the theorem using geometric illustrations and introduces the concept of Pythagorean triples. Practical problems, such as calculating the length of a ladder and solving for unknowns in triangles, are solved using the theorem.

Takeaways

• 📐 **Pythagorean Theorem**: The video discusses the Pythagorean Theorem, a fundamental principle in mathematics that describes the relationship between the sides of a right triangle.
• 🔢 **Triangle Classification**: It explains how to classify triangles based on the lengths of their sides, into obtuse, right, or acute triangles.
• 📚 **Concept of Triangles**: The script introduces the concept of triangles based on their angles, dividing them into three categories: acute, right, and obtuse.
• 📐 **Application of Pythagorean Theorem**: The video demonstrates how to apply the Pythagorean Theorem to determine the type of triangle given the lengths of its sides.
• 📐 **Example Calculation**: It provides an example of calculating whether a triangle with sides 2, 3, and 4 is obtuse, right, or acute.
• 📐 **Proof of Pythagorean Theorem**: The script includes a visual proof of the Pythagorean Theorem using a square divided into four right triangles and rearranged to form two squares.
• 🔢 **Triple Pythagoras**: It introduces the concept of Triple Pythagoras, which are sets of three numbers that satisfy the Pythagorean Theorem.
• 📚 **Verification of Triple Pythagoras**: The video shows how to verify a set of numbers as a Triple Pythagoras by checking if they satisfy the theorem.
• 📐 **Practical Application**: It demonstrates the practical application of the Pythagorean Theorem to solve real-world problems, such as finding the length of a ladder.
• 🔢 **Problem Solving**: The script includes problem-solving exercises that apply the Pythagorean Theorem to find unknown side lengths in triangles.
• 📚 **Final Problem**: The video concludes with a complex problem that uses the Pythagorean Theorem to find the value of 'Q' in a geometric setup.

Q & A

What is the main topic discussed in the video?

The main topic discussed in the video is the Pythagorean Theorem, specifically for 8th-grade middle school mathematics.

What are the three types of triangles based on the size of their angles?

The three types of triangles based on the size of their angles are acute, right, and obtuse triangles.

What is the condition for a triangle to be an acute triangle?

A triangle is an acute triangle if the square of its largest angle (C) is less than the sum of the squares of the other two sides (a and b).

How is a right triangle defined according to the Pythagorean Theorem?

A right triangle is defined as having the square of the length of its hypotenuse (C) equal to the sum of the squares of the lengths of the other two sides (a and b).

What is the condition for a triangle to be an obtuse triangle?

An obtuse triangle is one where the square of its largest angle (C) is greater than the sum of the squares of the other two sides (a and b).

What is the Pythagorean Theorem?

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

How is the Pythagorean Theorem proven in the video?

The Pythagorean Theorem is proven in the video by illustrating how four identical right triangles can be arranged to form a square, with the area of the square being equal to the sum of the areas of the individual triangles.

What is a Pythagorean Triple?

A Pythagorean Triple consists of three positive integers that satisfy the Pythagorean Theorem, meaning that the sum of the squares of the two smaller integers is equal to the square of the largest integer.

How is the Pythagorean Theorem applied in the second exercise of the video?

In the second exercise, the Pythagorean Theorem is applied to calculate the length of a ladder leaning against a wall, given the height of the wall and the distance from the wall to the foot of the ladder.

What is the solution to the third problem in the video involving the value of 'a'?

The solution to the third problem involves rearranging the Pythagorean Theorem to solve for 'a', which results in 'a' being equal to 12 after simplifying the equation.

How is the value of 'Q' determined in the final problem of the video?

The value of 'Q' in the final problem is determined by applying the Pythagorean Theorem to a series of triangles and solving the resulting equation, which leads to 'Q' being equal to 12 cm.

Outlines

00:00

📐 Introduction to Pythagorean Theorem

This paragraph introduces the Pythagorean Theorem, a mathematical concept taught in the 8th grade of middle school. It explains the different types of triangles based on their angles: acute, right, and obtuse. The theorem is described as a statement that must be proven and is attributed to the Greek mathematician Pythagoras. The theorem relates the lengths of the sides of a right triangle, stating that the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse. An example is given to illustrate how to determine if a triangle with sides of lengths 2, 3, and 4 is acute, right, or obtuse by comparing the square of the longest side to the sum of the squares of the other two sides.

05:04

🔍 Practical Applications of the Pythagorean Theorem

The second paragraph delves into practical applications of the Pythagorean Theorem. It presents a problem involving a triangle with sides of 10 cm and 14 cm, and an unknown side 'c', with the perimeter being 39 cm. By using the theorem, the length of side 'c' is calculated to be 15 cm. The paragraph then discusses how to determine the type of triangle by comparing the square of the longest side to the sum of the squares of the other two sides. Another problem involves calculating the length of a ladder leaning against a wall, given the height of the wall and the distance from the wall to the foot of the ladder. The Pythagorean Theorem is used to find the ladder's length, which is determined to be 15 meters. The paragraph also includes a problem-solving approach to find the value of 'a' in a right triangle where 'b' and 'c' are given, using algebraic manipulation.

10:05

🧩 Solving Complex Pythagorean Problems

The final paragraph focuses on solving more complex problems using the Pythagorean Theorem. It introduces the concept of Pythagorean triples, which are sets of three positive integers that satisfy the theorem. The paragraph presents a problem involving a larger triangle composed of smaller triangles, where the values of 'Q', 'P', and 'R' need to be determined. By setting up equations based on the theorem and solving them, the value of 'Q' is found to be 12 cm. The explanation involves combining equations and simplifying them to find the unknown values, demonstrating a deeper understanding of the theorem's application in solving geometric problems.

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. The video provides a proof of the theorem using a geometric illustration involving a square and four identical right-angled triangles.

💡Hypotenuse

The hypotenuse is the longest side of a right-angled triangle, opposite the right angle. In the context of the video, the hypotenuse is denoted by 'c' and is used in the Pythagorean Theorem formula (a² + b² = c²). The script uses the term when explaining the theorem and in solving practical problems, such as calculating the length of a ladder.

💡Acute Triangle

An acute triangle is a type of triangle where all three angles are less than 90 degrees. The video script mentions that if the square of the hypotenuse (c²) is less than the sum of the squares of the other two sides (a² + b²), then the triangle is acute. This concept is used to categorize triangles based on the relationship between their side lengths.

💡Right Triangle

A right triangle is a triangle that has one angle that is exactly 90 degrees. The Pythagorean Theorem specifically applies to right triangles, as it relates the lengths of the triangle's sides. The video uses the term 'right triangle' when explaining the theorem and in the context of the theorem's proof.

💡Obtuse Triangle

An obtuse triangle is one in which one of the angles is greater than 90 degrees. The video explains that if c² is greater than a² + b², then the triangle is obtuse. This term is crucial for understanding the classification of triangles and the conditions under which the Pythagorean Theorem holds true.

💡Square

In the video, 'square' refers to the mathematical operation of multiplying a number by itself, which is used extensively in the Pythagorean Theorem. The script mentions 'square' when explaining the theorem (a², b², c²) and when calculating the area of shapes in the geometric proof of the theorem.

💡Proof

A proof in mathematics is a logical argument for the truth of a proposition. The video provides a proof of the Pythagorean Theorem using a geometric construction involving a square and triangles. The term 'proof' is essential to the video's narrative as it demonstrates the validity of the theorem through a visual, step-by-step argument.

💡Illustration

An illustration in the context of the video refers to a visual representation or diagram used to explain or prove a concept. The script mentions an 'illustration' when describing the geometric proof of the Pythagorean Theorem, which helps viewers understand the relationship between the areas of squares and triangles.

💡Triangle Classification

The video script discusses classifying triangles based on the angles they contain: acute, right, and obtuse. This classification is essential for understanding the conditions under which the Pythagorean Theorem applies and for solving problems involving triangles.

💡Exercise

The term 'exercise' in the video refers to the practice problems that are presented to apply the concepts discussed, such as using the Pythagorean Theorem to find the length of a triangle's sides or the length of a ladder. These exercises are used to reinforce the understanding of the theorem and its practical applications.

💡Subscription

In the video script, 'subscription' refers to the act of subscribing to the channel, which is a common call-to-action for content creators to grow their audience. The term is used at the beginning and end of the video, highlighting its importance in engaging viewers and promoting the channel.

Highlights

Introduction to Pythagorean Theorem for 8th-grade SMP students.

Explanation of triangle types based on angle sizes: acute, right, and obtuse.

Criteria for identifying triangle types using the Pythagorean theorem.

Example calculation to determine if a triangle with sides 2, 3, and 4 is acute, right, or obtuse.

Concept of the Pythagorean theorem as a statement that must be proven.

Historical context of Pythagoras, the Greek mathematician who discovered the theorem.

Explanation of the relationship between the sides of a right triangle in the Pythagorean theorem.

Illustration of proving the Pythagorean theorem using a square.

Introduction to Triple Pythagoras, sets of three numbers that satisfy the theorem.

Verification of a Triple Pythagoras set (5, 12, 13) using the theorem.

Practical exercise problem involving the sides of a triangle with lengths 10 cm, 14 cm, and unknown c.

Calculation to determine the type of triangle based on the sides' lengths.

Problem-solving approach for a ladder leaning against a wall using the Pythagorean theorem.

Detailed calculation to find the length of a ladder based on the height and distance from the wall.

Exercise problem involving calculating the value of 'a' in a right triangle with given side lengths.

Step-by-step algebraic manipulation to solve for 'a' using the Pythagorean theorem.

Final exercise problem involving calculating the value of 'Q' in a complex right triangle setup.

Use of the Pythagorean theorem to solve for 'Q' by setting up and solving equations.

Conclusion of the video with a reminder to subscribe, like, and share.

Transcripts

Assalamualaikum Welcome back Selamat  datang kembali di matematika Mania  

pada video kali ini kita akan membahas  materi kelas 8 SMP teorema Pythagoras  

sebelum kita lanjut ke pembahasan jangan lupa  subscribe like and share ya terima kasih [Musik]

konsep segitiga berdasarkan besar  sudutnya segitiga dibagi menjadi tiga  

yaitu segitiga lancip segitiga  siku-siku dan segitiga tumpul

jika ada segitiga ABC dengan panjang sisi a  b c maka berlaku ketentuan sebagai berikut  

apabila C kuadrat kurang dari a kuadrat ditambah  b kuadrat maka segitiga ABC tersebut merupakan  

segitiga lancip jika C kuadrat sama dengan a² +  b² maka segitiga ABC merupakan segitiga siku-siku  

kemudian jika C kuadrat lebih dari a kuadrat +  b² maka segitiga ABC merupakan segitiga tumpul  

perhatikan contoh berikut misal ada segitiga  dengan panjang sisi 2 3 dan 4 maka kita akan  

cek segitiga tersebut apakah termasuk Lancip  siku-siku atau tumpul maka nilai C kuadrat kita  

hitung sama dengan 4 kuadrat = 16 kemudian a² + b  kuadrat kita hitung sama dengan 2 kuadrat ditambah  

3 kuadrat = 4 + 9 yaitu 13 artinya adalah c²  yaitu 16 lebih besar dari a² + b kuadrat yaitu 13  

maka segitiga tersebut merupakan segitiga tumpul

konsep teorema Pythagoras

teorema yaitu suatu pernyataan  yang harus dibuktikan kebenarannya  

kemudian pythagoras adalah matematikawan yang  berasal dari Yunani yang menemukan hubungan  

panjang sisi-sisi pada segitiga siku-siku teorema  Pythagoras berisi tentang pada segitiga siku-siku  

berlaku jumlah kuadrat dari panjang sisi-sisi  tegak sama dengan kuadrat panjang sisi miringnya

misalkan ada segitiga siku-siku sebagai berikut  maka berdasarkan teorema tersebut bisa kita  

simpulkan a² + b² = C kuadrat sisi miring yaitu  Sisi yang berhadapan dengan sudut siku-siku  

kita akan buktikan teorema Pythagoras  dalam sebuah ilustrasi berikut  

misalkan kita mempunyai segitiga seperti  ini kemudian kita mempunyai sebuah persegi

kita masukkan segitiga ABC  tersebut ke dalam persegi  

kemudian kita gandakan segitiga  tersebut sebanyak 4 buah dengan  

ukuran yang sama maka akan terbentuk seperti  ini kita perhatikan yang berwarna abu-abu

ini adalah bentuk persegi dengan Sisinya adalah C  maka kita bisa menghitung luasnya yaitu C kuadrat

gambar ini kita gandakan

kemudian segitiga ini kita geser sedemikian rupa

Maka kalau kita perhatikan akan muncul dua persegi  

dimana persegi kecil dengan Sisi a maka  mempunyai luas a² kemudian persegi besar  

dengan Sisi b maka Luasnya sama dengan b kuadrat  persegi sebelah kiri dan sebelah kanan mempunyai  

ukuran yang sama untuk yang warna abu-abu  sehingga terbukti bahwa a kuadrat + b² = C kuadrat

Triple Pythagoras adalah 3 pasang  bilangan yang memenuhi teorema Pythagoras

Apabila kita mempunyai segitiga siku-siku  sebagai berikut maka Tripel pythagorasnya adalah

untuk membuktikan kita akan cek salah satu  Triple Pythagoras tersebut misalnya yang ini  

maka a² + b² = c² kita masukkan 5  kuadrat ditambah 12 kuadrat = 13²  

5² = 25 12 kuadrat = 144 kemudian 13² =  169 maka setelah kita jumlahkan 169 = 169  

sehingga terbukti latihan soal teorema Pythagoras  nomor 1 diketahui sisi-sisi segitiga adalah 10 cm  

14 cm dan c cm jika keliling segitiga tersebut  adalah 39 cm maka jenis segitiga tersebut adalah

hanya 10 kemudian b nya 14  maka c nya bisa kita cari

diketahui sekelilingnya adalah  39 cm maka a + b + c = 39 maka  

10 + 14 + C = 39 maka 24 + C = 39  sehingga C = 39 dikurangi 24 = 15

setelah kita mengetahui nilai ABC  maka kita akan cek jenis segitiga  

tersebut dengan menghitung C kuadrat = 15 kuadrat  

yaitu 225 kemudian kita hitung a² + b² =  10² + 14² maka a² + b² = 100 + 196 = 296

maka kita bandingkan nilai  C kuadrat dan a² + b kuadrat  

karena C kuadrat kurang dari a² + P kuadrat  maka segitiga tersebut merupakan segitiga lancip

latihan soal nomor 2 sebuah tangga bersandar  pada dinding rumah yang tingginya 9 m

jika jarak kaki tangga dan dinding 12 M  maka berapa meter panjang tangga tersebut  

kita ilustrasikan sebagai berikut

untuk menghitung panjang tangga tersebut kita bisa  menggunakan teorema Pythagoras dimana rumusnya  

adalah a kuadrat + b² = C kuadrat kita masukkan  nilai-nilainya 9 kuadrat ditambah 12 kuadrat = c²  

maka 81 + 144 = c² sehingga c² = 225 sehingga  kita bisa menghitung nilai C = akar 225 = 15

jadi panjang tangga tersebut adalah 15 m

soal nomor 3 Perhatikan gambar  berikut nilai a yang memenuhi adalah

untuk menghitung nilai a maka kita akan  menggunakan teorema Pythagoras dimana a  

kuadrat + b² = C kuadrat kemudian kita masukkan  nilai-nilainya a-nya = a² kemudian b nya 16  

kemudian c nya adalah a + 8 maka kita  jabarkan a kuadrat + 256 = a² + 16 a + 64  

kemudian kita gabungkan kita geser a² dan 64 ke  kiri maka menjadi a² - a² + 256 dikurangi 64 = 16a  

karena a² berarti menjadi 0 kemudian 256 -  64 = 192 maka 192 = 16a maka nilai a bisa  

kita hitung yaitu 192 dibagi dengan 16 maka  a = 12 jadi nilai a yang memenuhi adalah 12

soal terakhir Perhatikan gambar berikut  Berapakah nilai Q untuk menyelesaikan  

soal ini kita masih akan menggunakan teorema  Pythagoras dimana a kuadrat + b² = C kuadrat  

perhatikan segitiga yang ini maka  kita masukkan nilainya Q kuadrat  

ditambah dengan 9 kuadrat = P kuadrat  kemudian perhatikan segitiga yang ini  

maka q kuadrat ditambah dengan 16 kuadrat = r  kuadrat sekarang perhatikan segitiga yang besar  

maka kita mempunyai persamaan P kuadrat ditambah r  kuadrat = 25 kuadrat 25 berasal dari 9 ditambah 16

tadi kita mempunyai persamaan P  kuadrat kita mempunyai persamaan  

r² maka persamaan ini kita masukkan ke  dalam persamaan ketiga sehingga menjadi  

seperti ini q² + 9 kuadrat ditambah  I kuadrat ditambah 16 kuadrat = 25²

maka kita gabungkan nilai  yang sama q² dengan q² = 2q²  

dan ditambah 81 + 256 = 337 maka 2q² =  625 - 337 sehingga 2q² = 288 maka q² = 144

Q = √144 maka Q = 12

jadi nilai Q adalah 12 cm

Oke Cukup Sekian dulu ya untuk pembahasan kali  ini Sampai ketemu di video-video selanjutnya  

jangan lupa subscribe like and share  ya terima kasih Assalamualaikum bye bye