recalcule sabendo-se que o lado maior do quadrilátero é 180 e o lado menor é 120

Let's recalculate given that the larger side of the rectangle (quadrilateral) is 180 and the smaller side is 120. The goal is still to find the angle \( B \) for which the diagonal of the rectangle equals 15.

### Step-by-Step Solution

1. **Given Dimensions**:
   - Larger side of the rectangle \( x = 180 \)
   - Smaller side of the rectangle \( y = 120 \)
   - Diagonal (hypotenuse of the triangle) \( d = 15 \)

2. **Right Triangle Formation**:
   - The rectangle forms a right triangle with the diagonal as the hypotenuse, and sides of lengths 120 and 180.

3. **Using the Pythagorean Theorem**:
   - For a right triangle with legs \( 120 \) and \( 180 \), and hypotenuse \( d = 15 \), the Pythagorean theorem states:
   \[
   d^2 = x^2 + y^2
   \]
   Plugging in the values:
   \[
   15^2 = 180^2 + 120^2
   \]

4. **Calculation**:
   Let's calculate the values:
   \[
   15^2 = 225
   \]
   \[
   180^2 = 32400
   \]
   \[
   120^2 = 14400
   \]
   Substituting these into the Pythagorean equation:
   \[
   225 = 32400 + 14400 = 46800
   \]

   We notice that the Pythagorean theorem doesn’t hold with these numbers, meaning the given hypotenuse (15) is incorrect under these dimensions for a rectangle.

### Correct Approach

Since the diagonal of a rectangle cannot be shorter than any of its sides, there must be an error in interpreting the problem. Given the side lengths of 120 and 180, the diagonal cannot be 15. Let's instead find what the actual diagonal should be.

5. **Correct Diagonal Calculation**:
   - The actual diagonal of the rectangle with side lengths 120 and 180 can be calculated using:
   \[
   d = \sqrt{120^2 + 180^2}
   \]
   \[
   d = \sqrt{14400 + 32400}
   \]
   \[
   d = \sqrt{46800}
   \]
   \[
   d = 216.33 \text{ (approximately)}
   \]

### Finding Angle \( B \)

To find the angle \( B \) that the diagonal forms with one of the sides (let's assume the smaller side, 120), we can use the trigonometric function tangent:

\[
\tan(B) = \frac{\text{opposite side}}{\text{adjacent side}} = \frac{180}{120} = 1.5
\]

Then,

\[
B = \tan^{-1}(1.5)
\]

Calculating this:

\[
B \approx 56.31^\circ
\]

### Conclusion

With the side lengths of 120 and 180, the diagonal is approximately 216.33, and the angle \( B \) is approximately \( 56.31^\circ \).

Would you like further details or have any questions?

Here are some related questions for further exploration:
1. How do you calculate the length of the diagonal of any rectangle given its sides?
2. What happens to the angle \( B \) if one side of the rectangle is doubled?
3. Can the Pythagorean theorem be used in three-dimensional geometry?
4. What are other ways to calculate angles in a rectangle besides using the tangent function?
5. How does the concept of similar triangles apply to rectangles?

**Tip:** Always double-check if the given dimensions make sense with the properties of the geometric shape you're working with.

Learn how to recalculate the diagonal of a rectangle with sides measuring 120 and 180 using the Pythagorean theorem and trigonometric functions. This detailed solution covers the relationship between side lengths and diagonal length in geometric shapes. Suitable for high school students.

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