The problem provided is in French, and it states:

"A right triangle has a hypotenuse of 15 cm, a perimeter of 36 cm, and an area of 54 cm². Determine the lengths of the sides of the right angle."

Let's denote the sides of the triangle as follows:

• $a$ and $b$: the legs of the right triangle.
• $c = 15$: the hypotenuse.

We are given three pieces of information:

1. $c = 15$ cm (hypotenuse).
2. The perimeter $P = a + b + c = 36$ cm.
3. The area $A = \frac{1}{2} \times a \times b = 54$ cm².

We can solve this by using the system of equations based on the perimeter and area of the triangle, along with the Pythagorean theorem:

1. Perimeter equation: $a + b + 15 = 36 \quad \Rightarrow \quad a + b = 21$

2. Area equation: $\frac{1}{2} \times a \times b = 54 \quad \Rightarrow \quad a \times b = 108$

3. Pythagorean theorem: $a^2 + b^2 = 15^2 = 225$

We now have the system:

• $a + b = 21$
• $a \times b = 108$
• $a^2 + b^2 = 225$

I will solve this system of equations to find the values of $a$ and $b$.The solutions for the lengths of the two legs $a$ and $b$ of the right triangle are:

• $a = 9$ cm and $b = 12$ cm, or
• $a = 12$ cm and $b = 9$ cm.

Thus, the legs of the right triangle are 9 cm and 12 cm.

Would you like further details, or do you have any questions?

Here are 5 related questions:

1. How is the Pythagorean theorem applied to right triangles?
2. Can the perimeter formula be used for other types of triangles?
3. What happens to the area formula for a right triangle if only one leg is known?
4. How would the solution change if the perimeter was different but the hypotenuse remained the same?
5. How can the system of equations approach be used for solving other types of geometry problems?

Tip: For solving triangles, always use a combination of geometric formulas (such as perimeter, area, and Pythagoras) to form a system of equations, especially in problems with missing side lengths.