The uploaded document focuses on using the Pythagorean theorem to determine the nature of triangles and to calculate lengths in a right triangle.

Here is a summary:

Key Concepts:

1. Property 1: Determining if a Triangle is a Right Triangle

   • A triangle is right if the square of the longest side equals the sum of the squares of the two other sides.
   • $c^2 = a^2 + b^2$ (Pythagorean Theorem).
   • Example calculations confirm whether a triangle is right-angled.

2. Rappel de Calcul (Calculation Reminder):

   • To find the square of a number, calculate $x \times x$.
   • To find a square root, use the $\sqrt{x}$ function.

3. Property 2: Calculating a Side Length

   • In a right triangle, to find a missing side length:
     • Hypotenuse: $c = \sqrt{a^2 + b^2}$
     • Leg: $a = \sqrt{c^2 - b^2}$

Would you like a detailed solution or explanation for a specific part of this? Let me know if you have a particular problem in mind!

Here are 5 further exploration questions for better understanding:

1. Can you verify the nature of other triangles using the Pythagorean Theorem?
2. How do you find the hypotenuse in different triangles?
3. What happens if $c^2 < a^2 + b^2$? Is it still a right triangle?
4. Can you generalize these methods to 3D geometry?
5. How do we apply these calculations in real-world problems?

Tip: Always re-check which side is the longest before applying the Pythagorean Theorem!