MATEMATIKA Kelas 8 - Teorema Phytagoras | GIA Academy
Summary
TLDRThis educational video from GYA Academy introduces viewers to the Pythagorean theorem, a fundamental principle in mathematics for calculating the length of the hypotenuse in a right-angled triangle. The video explains the theorem using a real-life scenario of estimating the height of a person on a skyscraper. It demonstrates how to apply the theorem with a step-by-step example and introduces the concept of Pythagorean triples for quicker calculations. The video also covers how to determine the type of triangle based on side lengths and encourages practice with provided problems to solidify understanding.
Takeaways
- 🏙️ The script introduces a scenario of measuring the distance to a person standing on top of a skyscraper, highlighting the impracticality of using a meter stick for such a tall building.
- 📏 It suggests that measuring such distances requires a more sophisticated approach, hinting at the use of mathematical theories rather than simple measurement tools.
- 🧮 The script references Pythagoras, a Greek mathematician and philosopher, who is renowned for his famous theorem that relates the sides of a right-angled triangle.
- 🔢 Pythagorean theorem is introduced as a formula a^2 = b^2 + c^2, where a is the hypotenuse and b and c are the other sides of the triangle.
- 📐 The script explains the application of the Pythagorean theorem to calculate the missing side of a triangle when the other two sides are known.
- 📊 An example problem is presented where the hypotenuse is 15 cm and one leg is 9 cm, guiding through the calculation to find the missing side.
- 🔄 The concept of Pythagorean triples is introduced as a quick way to determine the sides of a right-angled triangle without complex calculations.
- 📋 A table of Pythagorean triples is mentioned, which can be used to instantly identify the sides of a right-angled triangle if the numbers match those in the table.
- 🔍 The script discusses how to determine the type of triangle (right-angled, obtuse, or acute) based on the lengths of its sides using Pythagorean theorem.
- 🏛️ Finally, the script connects the mathematical discussion back to the initial scenario, explaining how the Pythagorean theorem can be used to calculate the distance to a person on top of a skyscraper.
Q & A
Who is Pythagoras and what is his significant contribution to mathematics?
-Pythagoras was a Greek mathematician and philosopher born on the island of Samos around 570 BCE. His most famous contribution to mathematics is the Pythagorean theorem, which describes the relationship between the sides of a right-angled triangle.
What does the Pythagorean theorem state?
-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.
What is the formula for the Pythagorean theorem?
-The formula for the Pythagorean theorem is a² = b² + c², where a is the length of the hypotenuse, and b and c are the lengths of the other two sides.
How can the Pythagorean theorem be used to determine the height of a building?
-The Pythagorean theorem can be used to determine the height of a building by measuring the distance from the observer to the base of the building and the distance from the observer to the top of the building. These measurements can then be used to calculate the height using the theorem.
What is a Pythagorean triple?
-A Pythagorean triple consists of three positive integers that satisfy the Pythagorean theorem. The largest number represents the hypotenuse, and the other two numbers represent the lengths of the other two sides of a right-angled triangle.
How can one quickly find the missing side of a right-angled triangle using Pythagorean triples?
-By identifying the known sides as part of a known Pythagorean triple, one can quickly determine the missing side without complex calculations. The largest number in the triple is the hypotenuse, and the other two numbers are the other two sides of the triangle.
How can the Pythagorean theorem help in identifying the type of triangle based on its side lengths?
-The Pythagorean theorem can help identify if a triangle is right-angled, obtuse, or acute by comparing the square of the longest side to the sum of the squares of the other two sides. If the longest side's square equals the sum, it's a right-angled triangle; if it's greater, it's an obtuse triangle; if it's less, it's an acute triangle.
What is the significance of the term 'acute' when classifying triangles?
-An acute triangle is one where all angles are less than 90 degrees. Using the Pythagorean theorem, if the square of the longest side is less than the sum of the squares of the other two sides, the triangle is classified as acute.
What is the significance of the term 'obtuse' when classifying triangles?
-An obtuse triangle is one where one angle is greater than 90 degrees. According to the Pythagorean theorem, if the square of the longest side is greater than the sum of the squares of the other two sides, the triangle is classified as obtuse.
How can the Pythagorean theorem be applied in real-life scenarios such as determining distances from a high vantage point?
-The Pythagorean theorem can be applied to calculate the distance to an object at a high vantage point by using the height of the vantage point and the horizontal distance to the base of the object. This allows for the calculation of the vertical distance to the object.
Outlines
📏 Introduction to Pythagorean Theorem
This paragraph introduces the concept of the Pythagorean Theorem through a relatable scenario of estimating the distance to someone on top of a tall building. It explains that measuring such distances with a meter stick is impractical due to the height. The paragraph then introduces Pythagoras, a Greek mathematician and philosopher, and his famous theorem which relates the sides of a right-angled triangle. The theorem is mathematically expressed as a^2 = b^2 + c^2, where a is the hypotenuse, and b and c are the other two sides. The explanation continues with an example problem where the hypotenuse is 15 cm and one leg is 9 cm, and the goal is to find the length of the other leg using the theorem.
🔢 Pythagorean Triples and Triangle Classification
The second paragraph delves into Pythagorean Triples, which are sets of three whole numbers that satisfy the Pythagorean Theorem. It mentions that these triples can be used to quickly determine the length of one side of a right-angled triangle when the other two are known. The paragraph provides an example using the numbers 9, 12, and 15, which are a Pythagorean Triple, to illustrate how one can easily find the missing side without complex calculations. It also discusses how to classify triangles based on the lengths of their sides using the theorem, distinguishing between acute, right, and obtuse triangles. The paragraph concludes with another example problem involving sides of 12 cm, 16 cm, and 19 cm, demonstrating how to determine the type of triangle using the theorem.
Mindmap
Keywords
💡Pythagoras
💡Pythagorean Theorem
💡Hypotenuse
💡Right Triangle
💡Sides of a Triangle
💡Square
💡Triple Pythagorean
💡Obtuse Triangle
💡Acute Triangle
💡Height of a Building
Highlights
Introduction to GYA Academy YouTube channel and a question about measuring distance.
Discussion on the ineffectiveness of using a meter stick to measure the distance to someone on a skyscraper.
Introducing Pythagoras' theorem as a method to determine distances.
Historical background of Pythagoras, the Greek mathematician and philosopher.
Explanation of Pythagoras' theorem relating the sides of a right-angled triangle.
Mathematical formulation of Pythagoras' theorem: a^2 = b^2 + c^2.
Clarification on the naming convention of triangle sides based on angles.
Example problem solving using Pythagoras' theorem to find the missing side of a triangle.
Step-by-step calculation to find the length of the missing side (AC) in the example.
Introduction of Pythagorean triples for quick calculation of triangle sides.
Definition of Pythagorean triples and their significance in simplifying calculations.
Table of Pythagorean triples provided for reference.
Using Pythagorean triples to identify the missing side without complex calculations.
Determining the type of triangle based on the lengths of its sides.
Method to determine if a triangle is acute, right, or obtuse using Pythagorean theorem.
Example of determining the type of triangle using the lengths of its sides.
Conclusion on using Pythagoras' theorem to measure distances and the practicality of the method.
Encouragement to practice problems for better understanding and to continue watching the channel.
Transcripts
Hai Cinta
[Musik]
hai hai
halo halo teman teman selamat datang
kembali di GYA Academy youtube channel
Oh iya apa kabar semoga kita sehat
selalu ya Pernahkah teman-teman berada
di dekat sebuah gedung pencakar langit
lalu di atas gedung terlihat seseorang
yang sedang menikmati pemandangan dari
ketinggian tersebut kira-kira berapa ya
jarak kita dengan orang di atas gedung
teman-teman ada yang tahu cara
menentukannya
hai hai
Hai mungkin mengukur menggunakan meteran
adalah cara yang terlintas di pikiran
sebagian kita tapi gedung tersebut
sangat tinggi sehingga jaraknya sangat
jauh Apakah mengukur menggunakan meteran
efektif kita gunakan pada kondisi ini
tentu tidak
[Musik]
Hai jadi teman-teman untuk mengetahui
jarak kita dengan orang di atas gedung
bisa menggunakan teori yang dikemukakan
oleh seorang ilmuwan matematika yang
bernama vitago ras pythagoras merupakan
seorang matematikawan dan filsuf Yunani
yang lahir pada tahun 570 sebelum masehi
di Pulau samos Banyak sekali penemuan
pythagoras sehingga beliau diberi
julukan sebagai Bapak bilangan namun
penemuan pythagoras yang sangat terkenal
adalah teorema Pythagoras
Hai teorema Pythagoras adalah sebuah
teori yang menunjukkan hubungan panjang
sisi-sisi pada segitiga siku-siku nah
kuadrat panjang sisi miring suatu
segitiga siku-siku ini merupakan jumlah
kuadrat panjang sisi-sisi lainnya yaitu
sisi alas dan Sisi tegak segitiga
tersebut
Hai secara matematis teorema Pythagoras
dapat ditulis dengan persamaan a kuadrat
= b kuadrat ditambah y kuadrat dengan
aadalah sisi miring segitiga sedangkan b
dan c adalah sisi segitiga yang lainnya
Hai pada segitiga penamaan Sisinya dapat
kita lakukan dengan memperhatikan sudut
di depan sisi tersebut Sisi yang berada
di depan sudut a kita sebut sisi-sisi di
depan sudut b adalah Sisi B dan Sisi
yang berada di depan sudut c adalah Sisi
C bisa dimengerti ya teman-teman
Sekarang kita akan membahas contoh soal
dari soal kita bisa perhatikan bahwa
sisi miring segitiga adalah 15 cm dan
Sisi alasnya 9 cm pada soal ini berarti
kita akan menentukan Sisi tegak segitiga
atau sisi AC untuk menyelesaikan soal
ini kita ingat lagi persamaan teorema
Pythagoras hal kuadrat = b kuadrat
ditambah y kuadrat untuk soal ini
menjadi becek kuadrat = AB kuadrat
ditambah Aceh kuadrat
Hai karena yang ditanya pada soal adalah
Sisi tegak yaitu Aceh maka sesuai aturan
pindah ruas pada operasi bilangan Aceh
kuadrat = BC kuadrat dikurang AB kuadrat
lalu kita masukkan angkanya BC = 15 cm
jangan lupa dikuadratkan ya ABS 9 cm
dikuadratkan 15 kuadrat = 225 dikurang 9
kuadrat = 81 sehingga Aceh kuadrat =
144°
Hai jika Aceh kuadrat = 144° kacamata21
drat yaitu akar 144 akar 144 = 12 jadi
jawabannya C12 cm
[Musik]
Hai nah teman-teman selain menggunakan
rumus phytagoras kita juga dapat
menggunakan Tripel pythagoras cara ini
bisa menemukan panjang Salah satu sisi
segitiga dengan lebih cepat Tripel
pythagoras merupakan kelompok tiga
bilangan asli yang memenuhi ketentuan
yaitu kuadrat bilangan terbesar sama
dengan jumlah kuadrat dua bilangan
lainnya jadi bilangan yang terbesar pada
Tripel pythagoras adalah sisi miring
segitiga
Hai berikut adalah tabel Tripel
pythagoras dari tabel kita bisa
mengetahui bilangan-bilangan yang
memenuhi syarat Tripel pythagoras
misalnya 345 dan kelipatannya jika kita
kembali pada contoh soal di atas dua
sisi yang diketahui adalah 15 dan 9 lalu
perhatikan kembali tabel Tripel
pythagoras dari tabel bilangan yang
memenuhi adalah 9 12 dan 15 jadi tanpa
menggunakan rumus dan perhitungan yang
mungkin cukup menguras waktu dan pikiran
kita bisa mengetahui panjang sisi yang
belum diketahui tersebut secara langsung
sepertinya cara ini lebih mudah ya
teman-teman
Hai Oke selanjutnya kita akan menentukan
jenis segitiga berdasarkan panjang
sisi-sisinya Bagaimana cara menentukan
jenis segitiga jika panjang ketiga
Sisinya diketahui jadi kita bisa
menentukannya menggunakan perhitungan
Tripel pythagoras yang pertama jika
aquadrat = b kuadrat ditambah y kuadrat
maka segitiga ABC siku-siku di a yang
kedua jika aquadrat besar dari b kuadrat
ditambah y kuadrat maka segitiga ABC
tumpul dia dan yang terakhir jika
aquadrat kecil dari b kuadrat ditambah y
kuadrat maka segitiga ABC Lancip dia
begitu ya teman-teman
Hai sekarang untuk lebih paham kita
bahas contoh soalnya yuk dari soal
diketahui sisi segitiga adalah 12 cm
ke-16 cm dan 19 cm agar kita dapat
mengetahui jenisnya kita buat
perhitungan untuk menentukan hubungan
sisi miring dengan dua sisi lainnya
misalkan a panjang sisi miring sedangkan
b&c panjang sisi yang lain a = 16 cm B =
12 cm dan Ce = 16 cm maka 19 kuadrat =
12 kuadrat ditambah 16 kuadrat 19
kuadrat = 361 12 kuadrat = 144° kuadrat
= 256 144 ditambah 256 sama dengan 400
karena 19 kuadrat lebih kecil dari 2
squadra ditambah 16 kuadrat maka
segitiga ini Lancip dia sekarang udah
terjawab ya pertanyaan di awal tadi jadi
dengan menggunakan teorema Pythagoras
kita dapat menentukan jarak kita dengan
orang yang berada di atas gedung
pastinya kita perlu tahu dulu jarak kita
terhadap gedung dan tinggi gedung diatas
kita
Hai Nah berarti kita nggak perlu cari
meteran ya kita cukup menghitung dengan
rumus atau menggunakan tabel Tripel
pythagoras Oke teman-teman semua bisa
dipahami ya Jangan lupa latihan soalnya
biar teman-teman lebih mengerti lagi
jangan lupa saksikan terus video video
di channel kita ya sampai jumpa
hai hai hai
hai hai
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