Understanding Parallelograms Formed by Parallel Lines

The question asks us to find the value of $(m + n)$ given that $m$ parallel lines intersecting $n$ parallel lines create 60 parallelograms. This is a common problem involving combinations in geometry.

Formula for Counting Parallelograms

When a set of $m$ parallel lines intersects another set of $n$ parallel lines, the number of parallelograms formed is determined by choosing two lines from the first set and two lines from the second set to form the sides of a parallelogram.

• To choose 2 lines from $m$ parallel lines, the number of ways is given by the combination formula $\binom{m}{2}$.
• To choose 2 lines from $n$ parallel lines, the number of ways is given by the combination formula $\binom{n}{2}$.

The total number of parallelograms is the product of these two combinations:

Number of parallelograms = $\binom{m}{2} \times \binom{n}{2}$

Setting up the Equation

We are given that the total number of parallelograms is 60. So, we can write the equation:

$\binom{m}{2} \times \binom{n}{2} = 60$

Solving the Equation

Recall the formula for combinations: $\binom{k}{2} = \frac{k(k-1)}{2}$.

Substituting this into our equation:

$\frac{m(m-1)}{2} \times \frac{n(n-1)}{2} = 60$

Multiply both sides by 4:

$m(m-1)n(n-1) = 240$

We need to find integer values for $m$ (where $m \ge 2$) and $n$ (where $n \ge 2$) such that the product of $m(m-1)$ and $n(n-1)$ is 240.

Let's look at the values of $k(k-1)$ for small integer values of $k$ ($k \ge 2$):

We need to find two numbers from the second column that multiply to 240. Let's check the factors of 240:

• 1 x 240 (240 is not in the list $k(k-1)$ for reasonable k)
• 2 x 120 (120 is not in the list)
• 3 x 80 (80 is not in the list)
• 4 x 60 (60 is not in the list)
• 5 x 48 (48 is not in the list)
• 6 x 40 (40 is not in the list)
• 8 x 30 (30 is in the list: 6x5. 8 is not $k(k-1)$)
• 10 x 24 (Neither 10 nor 24 are in the list)
• 12 x 20 (Both 12 and 20 are in the list!)

We found a pair: 12 and 20.

• If $m(m-1) = 12$, then $m = 4$ (since 4 x 3 = 12).
• If $n(n-1) = 20$, then $n = 5$ (since 5 x 4 = 20).

Alternatively, it could be $m=5$ and $n=4$, as the formula is symmetric with respect to $m$ and $n$.

Let's verify with $m=4$ and $n=5$:

Number of parallelograms = $\binom{4}{2} \times \binom{5}{2} = \frac{4 \times 3}{2} \times \frac{5 \times 4}{2} = 6 \times 10 = 60$. This matches the given information.

Finding the Value of (m + n)

In either case ($m=4, n=5$ or $m=5, n=4$), the value of $(m + n)$ is:

$m + n = 4 + 5 = 9$

Conclusion

The value of $(m + n)$ is 9.

Revision Table: Counting Parallelograms

Additional Information: Geometric Combinations

This problem is an example of using combinations to count geometric shapes. Here are some related concepts:

• Lines formed by points: The number of lines formed by $n$ non-collinear points is $\binom{n}{2}$. If some points are collinear, you subtract the combinations of collinear points and add 1 for the single line they form.
• Triangles formed by points: The number of triangles formed by $n$ non-collinear points is $\binom{n}{3}$.
• Rectangles formed by grid lines: Similar to parallelograms, the number of rectangles in a grid formed by $m$ horizontal and $n$ vertical lines is $\binom{m}{2} \times \binom{n}{2}$. (Rectangles are a special case of parallelograms).

Understanding combinations is crucial for solving problems involving counting arrangements or selections of objects, including geometric figures formed by intersecting lines or sets of points.