The exercise is asking to calculate the height $SH$ of the attic (grenier) in René's house, based on the geometric properties of the roof. Here's how to solve the problem step-by-step:

1. Understanding the Problem

The roof is in the shape of an isosceles triangle:

• Base $AB = 9 \, \text{m}$,
• Both slanted sides $AS$ and $SB$ measure $5.6 \, \text{m}$,
• $SH$ is the vertical height of the triangle, which splits $AB$ into two equal halves at point $H$.

This forms two right triangles $\triangle SHA$ and $\triangle SHB$.

2. Using the Pythagorean Theorem

In one of the right triangles (e.g., $\triangle SHA$):

• Hypotenuse $AS = 5.6 \, \text{m}$,
• Half of the base $AH = \frac{AB}{2} = \frac{9}{2} = 4.5 \, \text{m}$,
• Height $SH$ is the unknown we need to calculate.

Using the Pythagorean theorem: $AS^2 = AH^2 + SH^2$ Substituting the known values: $5.6^2 = 4.5^2 + SH^2$

3. Solving for $SH$

First, calculate $5.6^2$ and $4.5^2$: $5.6^2 = 31.36, \quad 4.5^2 = 20.25$

Substitute these values into the equation: $31.36 = 20.25 + SH^2$

Simplify to find $SH^2$: $SH^2 = 31.36 - 20.25 = 11.11$

Take the square root of both sides: $SH = \sqrt{11.11} \approx 3.33 \, \text{m}$

4. Final Answer

The height $SH$ of the attic is approximately: $\boxed{3.33 \, \text{m}}$

Would you like further clarification or details? Here are five related questions to deepen your understanding:

1. How does the Pythagorean theorem apply to other roof designs or structures?
2. What are the implications of changing the base length $AB$ or the side lengths $AS$?
3. How would the solution differ if $AB$ were not split evenly by $SH$?
4. Can you calculate the angle $\angle SAH$ using trigonometric functions?
5. What are other practical uses of calculating heights in similar geometric problems?

Tip: Always double-check your calculations, especially when squaring or taking square roots, as small errors can lead to incorrect results!