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The problem given is about a ladder leaning against a wall, with the objective of determining whether the wall is vertical or not.

### Problem Breakdown:
- The ladder is **2.5 meters** long.
- The distance between the base of the ladder and the wall is **1.5 meters**.
- You are asked to determine if the wall is vertical.

To solve this, we can use the **Pythagorean theorem**, which is applicable when we have a right-angled triangle. Here, the ladder acts as the hypotenuse of a triangle, the distance between the base of the ladder and the wall is one side, and the height from the ground to the point where the ladder touches the wall would be the other side (i.e., the height of the wall up to the ladder's contact point).

The Pythagorean theorem states:

\[
a^2 + b^2 = c^2
\]

where:
- \(a\) is the distance from the base of the ladder to the wall (1.5 m),
- \(b\) is the height where the ladder touches the wall (which we will calculate),
- \(c\) is the length of the ladder (2.5 m).

### Steps to Solve:
1. Apply the Pythagorean theorem:
   \[
   (1.5)^2 + b^2 = (2.5)^2
   \]
   \[
   2.25 + b^2 = 6.25
   \]
   \[
   b^2 = 6.25 - 2.25 = 4
   \]
   \[
   b = \sqrt{4} = 2
   \]

2. The height of the ladder against the wall is **2 meters**.

### Conclusion:
For the wall to be vertical, the ladder would form a right triangle. Since the height from the ground to the point where the ladder touches the wall is exactly 2 meters, and the Pythagorean relationship holds, the wall is indeed vertical.

---

Would you like further details or have any questions about this solution? Here are some follow-up questions to consider:

1. What happens if the distance between the ladder and the wall changes? Would the wall still be vertical?
2. How would you solve this problem if the ladder length or distance were unknown?
3. Can the Pythagorean theorem be applied if the wall is not vertical? Why or why not?
4. What are other ways to determine if a wall is vertical besides using the Pythagorean theorem?
5. How would you calculate the angle between the ladder and the ground?

**Tip:** Always check if a right-angled triangle can be applied when you have lengths of two sides in problems involving ladders and walls.

A 2.5 m long ladder leans against the wall of a building. The distance between the wall and the base of the ladder is 1.5 m. Explain how you would determine whether the wall is vertical.

Solve a problem where a 2.5 m ladder leans against a wall, and the distance from the base to the wall is 1.5 m. Learn how to use the Pythagorean theorem to check if the wall is vertical by calculating the height at which the ladder touches the wall.

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