TEOREMA PYTHAGORAS : MATEMATIKA KELAS 8 SMP

Summary

This video provides a comprehensive lesson on the Pythagorean theorem for Grade 8 students, explaining how to calculate the sides of right triangles using the formula c² = a² + b². It covers identifying triangle types, solving for unknown sides, and using Pythagorean triples. Special right triangles (45°-45°-90° and 30°-60°-90°) are explained with practical examples. The video also demonstrates real-life applications, such as calculating the shortest travel distance, stair lengths, and diagonals in geometric shapes. Through step-by-step explanations and examples, viewers gain a clear understanding of the theorem’s principles and practical uses.

Takeaways

• 😀 The Pythagorean theorem is used to calculate the sides of a right triangle using the formula c² = a² + b², where c is the hypotenuse.
• 😀 To find an unknown side of a right triangle, you can rearrange the formula: a² = c² - b² or b² = c² - a².
• 😀 A triangle is right-angled if c² = a² + b², acute if c² < a² + b², and obtuse if c² > a² + b².
• 😀 Worked examples demonstrate applying the theorem to rectangles, triangles, and finding unknown sides, including using square roots for calculations.
• 😀 Pythagorean triples are sets of three integers that form the sides of a right triangle, e.g., (3, 4, 5), (5, 12, 13), and multiples like (6, 8, 10).
• 😀 Special right triangles include 45°-45°-90° triangles (isosceles) where hypotenuse = leg × √2.
• 😀 Another special triangle is 30°-60°-90°, where the hypotenuse = 2 × short side, and the long side = short side × √3.
• 😀 Real-life applications of the Pythagorean theorem include calculating the shortest path in navigation, ladder lengths, and diagonals of rectangles or 3D shapes.
• 😀 Example applications: Ladder to a window 8 m high across a 6 m garden → minimum ladder length = 10 m; a ship traveling 100 km west and 75 km south → shortest distance = 125 km.
• 😀 Understanding Pythagorean triples and special triangles helps simplify calculations without repeated use of the square root formula.
• 😀 Consistent labeling of triangle sides (a, b, c) and awareness of units (cm, m, km) is essential for accurate problem-solving.
• 😀 Repetition and step-by-step explanation in the script ensure students grasp both formulas and applications of the Pythagorean theorem.

Q & A

What is the Pythagorean theorem used for?

The Pythagorean theorem is used to calculate the sides of a right triangle.

What is the formula of the Pythagorean theorem?

The formula is c² = a² + b², where c is the hypotenuse and a and b are the other two sides of a right triangle.

How can you determine the type of triangle using the Pythagorean theorem?

If c² = a² + b², the triangle is right. If c² < a² + b², the triangle is acute. If c² > a² + b², the triangle is obtuse.

How do you calculate the width of a rectangle if the length and diagonal are known?

Use the formula width = √(diagonal² - length²). For example, if length = 16 cm and diagonal = 20 cm, width = √(400 - 256) = 12 cm.

What are Pythagorean triples and give examples?

Pythagorean triples are sets of three integers that form the sides of a right triangle. Examples include 3, 4, 5; 5, 12, 13; 7, 24, 25; and multiples such as 6, 8, 10.

What is special about a 45°-45°-90° triangle?

It is an isosceles right triangle where the two legs are equal and the hypotenuse is the leg length multiplied by √2.

How are the sides of a 30°-60°-90° triangle related?

In a 30°-60°-90° triangle, the short leg is a, the long leg is a√3, and the hypotenuse is 2a.

How can the Pythagorean theorem be applied in daily life?

It can be used to calculate the shortest distance between points, the length of stairs, and the diagonals of flat or spatial shapes.

If a ship travels 100 km west and then 75 km south, what is the shortest distance it can actually travel?

The shortest distance is the hypotenuse of the right triangle formed, calculated as √(100² + 75²) = 125 km.

How do you find a missing side of a right triangle if the hypotenuse and one leg are known?

Use the formula of the Pythagorean theorem: missing side = √(hypotenuse² - known leg²).

What is the method to find the value of x in a right triangle with sides 8x, 6x, and hypotenuse 20?

Use the Pythagorean theorem: (8x)² + (6x)² = 20² → 64x² + 36x² = 400 → 100x² = 400 → x = 2.

Why is it useful to memorize Pythagorean triples?

Memorizing Pythagorean triples allows for quick calculation of triangle sides without having to square and take roots repeatedly.