Two alternate sides of a regular polygon, when produced, meet at right angle. Calculate the number of sides in the polygon.

Answer

To solve the problem, we need to find the number of sides (n) in a regular polygon where two alternate sides, when extended, meet at a right angle (90 degrees).

### Step-by-Step Solution:

1. **Understand the Problem**: We know that when two alternate sides of a regular polygon are extended, they meet at a right angle. This means that the angle formed at the intersection is 90 degrees.

2. **Use the Interior Angle Formula**: The formula for the measure of each interior angle of a regular polygon with n sides is given by:
\[ \text{Interior Angle} = \frac{(n - 2) \times 180}{n} \]

3. **Set Up the Equation**: Since we know that the interior angle is equal to 90 degrees, we can set up the equation:
\[ \frac{(n - 2) \times 180}{n} = 90 \]

4. **Cross Multiply**: To eliminate the fraction, we can cross-multiply:
\[ (n - 2) \times 180 = 90n \]

5. **Distribute and Rearrange**: Distributing the 180 gives:
\[ 180n - 360 = 90n \]
Now, we rearrange the equation to isolate n:
\[ 180n - 90n = 360 \]

6. **Combine Like Terms**: This simplifies to:
\[ 90n = 360 \]

7. **Solve for n**: Dividing both sides by 90 gives:
\[ n = \frac{360}{90} = 4 \]

8. **Conclusion**: Therefore, the number of sides in the polygon is 4.

### Final Answer:
The number of sides in the polygon is **4**.

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