To solve this problem, let's go through the given information and apply geometric principles.

Given Information:

1. Height Difference (Peak Height): The roof peak (harja) is 1.8 meters higher than the top of the side walls.
2. Distance from Eaves to Wall: The perpendicular distance from the eave (räystäs) to the wall is 69 cm (0.69 m).
3. End Wall Length: The length of the end wall is 7.71 meters.
4. Tolerance Range: The answer must be within a tolerance of ±1 cm.
5. Objective: Find the length of the roof section $AB$.

Approach:

Since the peak height is in the center and the roof is symmetrical, we can use the following steps to find $AB$, the length of one side of the sloping roof:

1. Divide the End Wall in Half: Since the peak is in the center, each side of the end wall will be half of 7.71 m, which is: $\frac{7.71}{2} = 3.855 \text{ meters}$

2. Right Triangle Formation: Each side of the roof forms a right triangle where:

   • The horizontal leg is 3.855 meters.
   • The vertical leg (height difference from the wall to the peak) is 1.8 meters.

3. Calculate the Hypotenuse (AB): Using the Pythagorean theorem: $AB = \sqrt{(3.855)^2 + (1.8)^2}$

   Calculating the squares: $(3.855)^2 = 14.864025$ $(1.8)^2 = 3.24$

   Adding these: $AB = \sqrt{14.864025 + 3.24} = \sqrt{18.104025}$

   $AB \approx 4.26 \text{ meters}$

Answer:

The length of $AB$ is approximately 4.26 meters, with a tolerance of ±1 cm.

Would you like further clarification or have any questions?

Additional Questions to Explore:

1. What would be the length of $AB$ if the height difference changed?
2. How would the result change if the end wall length was different?
3. What if the eave-to-wall distance was different? How would that affect $AB$?
4. How does changing the tolerance range affect the precision of the answer?
5. Can we determine the roof angle using this information?

Tip:

When dealing with right triangles in construction problems, always double-check units (meters vs. centimeters) to maintain accuracy.