The Pythagorean Theorem and Right Triangles

This page covers the Pythagorean theorem, square roots, and methods for proving whether a triangle is right-angled. It provides essential information for students learning geometry and trigonometry.

1. The Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry, especially for right triangles.

Definition: In a right triangle, the square of the length of the hypotenuse is equal to the sum of squares of the lengths of the other two sides.

Example: In a right triangle ABC with right angle at A, the theorem states: AB² = AC² + CB²

Highlight: The hypotenuse is always the longest side of a right triangle, opposite the right angle.

2. Square Roots

Understanding square roots is crucial for applying the Pythagorean theorem.

Definition: The square root of a number is the value that, when multiplied by itself, equals the original number.

Example: √4 = 2 because 2² = 4, and √81 = 9 because 9² = 81

3. Proving a Triangle is Right-Angled

To démontrer qu'un triangle est rectangle, you need to apply the Pythagorean theorem.

Example: For triangle DEF with DE = 10, DF = 6, and FE = 8:

1. Identify the longest side (DE) as the potential hypotenuse
2. Calculate DE² = 10² = 100
3. Calculate FE² + DF² = 8² + 6² = 64 + 36 = 100
4. Since DE² = FE² + DF², the triangle is right-angled at F

Highlight: It's crucial to present calculations clearly to prouver qu'un triangle est rectangle avec 2 mesures.

4. Proving a Triangle is Not Right-Angled

To démontrer qu'un triangle est rectangle sans pythagore, you can show that the Pythagorean equation doesn't hold.

Example: For triangle GHI with GH = 11, HI = 4, and GI = 9:

1. Identify the longest side (GH) as the potential hypotenuse
2. Calculate GH² = 11² = 121
3. Calculate HI² + GI² = 4² + 9² = 16 + 81 = 97
4. Since GH² ≠ HI² + GI², the triangle is not right-angled

Highlight: This method is useful to comment démontrer qu'un triangle est rectangle sans mesure of angles.