Teorema Pythagoras Kelas 8 Semester 2
Summary
TLDRThis 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
📐 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.
🔍 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.
🧩 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.
Mindmap
Keywords
💡Pythagorean Theorem
💡Hypotenuse
💡Acute Triangle
💡Right Triangle
💡Obtuse Triangle
💡Square
💡Proof
💡Illustration
💡Triangle Classification
💡Exercise
💡Subscription
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
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