Question

A parallelogram is cut by two sets of m lines parallel to its sides. The number of parallelograms thus formed is ?
A.\[{\left( {{}^m{C_2}} \right)^2}\]
B.\[{\left( {{}^{m + 1}{C_2}} \right)^2}\]
C.\[{\left( {{}^{m + 2}{C_2}} \right)^2}\]
D.None of these

Answer 1

Hint: A parallelogram is a quadrilateral with two pairs of parallel sides. The opposite sides of a parallelogram are equal in length, and the opposite angles are equal in measure. Also, the interior angles on the same side of the transversal are supplementary. Sum of all the interior angles equals \[{360^ \circ }\].

Complete step-by-step answer:
The properties of a parallelogram are as follows:
The opposite sides are parallel and congruent
The opposite angles are congruent
The consecutive angles are supplementary
If anyone of the angles is a right angle, then all the other angles will be the right angle
The two diagonals bisect each other
Each diagonal bisects the parallelogram into two congruent triangles
Sum of squares of all the sides of the parallelogram is equal to the sum of squares of its diagonals. It is also called parallelogram law
The combination is a way of selecting items from a collection, such that (unlike permutations) the order of selection does not matter. In smaller cases, it is possible to count the number of combinations. Combination refers to the combination of n things taken k at a time without repetition. To refer to combinations in which repetition is allowed, the terms k-selection or k-combination with repetition are often used.
The two sets of m parallel lines along with two sets of two parallel lines of the given parallelogram will form two sets of \[m + 2\] parallel lines. To form a parallelogram we have to choose sets of \[2\] parallel lines from each of the above.
Hence \[\left( {{}^{m + 2}{C_2}} \right).\left( {{}^{m + 2}{C_2}} \right) = {\left( {{}^{m + 2}{C_2}} \right)^2}\] is the required number of parallelograms.
Therefore option (3) is the correct answer.
So, the correct answer is “Option 3”.

Note: The combination is a way of selecting items from a collection, such that (unlike permutations) the order of selection does not matter. A parallelogram is a quadrilateral with two pairs of parallel sides. The opposite sides of a parallelogram are equal in length, and the opposite angles are equal in measure.