Question

If $ m $ parallel lines are intersected by $ n $ parallel lines, then how many parallelograms are formed?

• A. \( m \times n \)
• B. \( \frac{m \times n}{4} \)
• C. \( \frac{m \times (m - 1) \times (n - 1)}{4} \)
• D. \( (m - 1) \times (n - 1) \)

Hint:

For problems involving intersections of parallel lines, remember that selecting two lines from each set forms a parallelogram.

Answer 1

The Correct Option is C

When \( m \) parallel lines are intersected by \( n \) parallel lines, parallelograms are formed by choosing two lines from the \( m \) lines and two lines from the \( n \) lines. The number of ways to select 2 lines from \( m \) lines is \( \binom{m}{2} \) and the number of ways to select 2 lines from \( n \) lines is \( \binom{n}{2} \). 
Thus, the total number of parallelograms formed is: \[ \binom{m}{2} \times \binom{n}{2} = \frac{m(m-1)}{2} \times \frac{n(n-1)}{2} \] Simplifying: \[ \frac{m(m-1) \times n(n-1)}{4} \] 
Thus, the correct answer is \( \frac{m \times (m - 1) \times (n - 1)}{4} \).