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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?

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### 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.

Talon harja on 1.8 metriä korkeammalla kuin sivuseinien yläreunat. Räystään kohtisuora etäisyys seinästä on 69 cm. Päätyseinän pituus on 7.71 m. Talon harja on keskellä kattoa. Määritä lappeen AB pituus. Vastauksen toleranssirajoiksi on asetettu ± 1 cm.

Learn how to calculate the length of roof slope AB for a symmetrical roof where the peak is 1.8 meters higher than the side walls and the end wall length is 7.71 meters. This problem uses the Pythagorean theorem to find AB with a tolerance of ±1 cm. Suitable for Grades 7-9.

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