The perimeter of square ABCD is half of the perimeter of parallelogram

The perimeter of square ABCD is half of the perimeter of parallelogram EFGH. What is the perimeter of quadrilateral EFGH?

(1) a line segment drawn from the center of ABCD to the midpoint of BC is 2 inches long

consider a square and from its center drop a line to side BC -- mid point of BC -- In a way it gives the side of the square as 4
hence perimeter is 16

given that perimeter of square = 1/2 parallelogram EFGH
therefore perimeter of EFGH = 32
Sufficient


(2) a line segment drawn from the center of EFGH to the midpoint of FG is 4 inches long
consider a parallelogram EFGH .... And follow the steps given in statement 2
We see that only side GH and EF can found out as 8 .
But what about other sides .

hence not sufficient .

A ans .

GROCKIT OFFICIAL SOLUTION:

As always with Data Sufficiency questions, we need to look at the statements individually first:

Statement 1: Because ABCD is a square, the distance from the center to any side’s midpoint is exactly half the distance of one side (and since it’s a square, all sides are the same). Since the line from center to BC’s midpoint is half a side length, BC itself is twice that distance.

2 x line segment = length of BC = 4

4 x BC = perimeter ABCD = 16

The perimeter of EFGH is twice that of ABCD, so perimeter of EFGH = 32. SUFFICIENT.

Statement 2: The statement tells us that the length of EG is 8 (twice the line segment parallel to it), but EFGH is a parallelogram, which means that it has four sides that are not necessarily the same length. If EFGH is also a square, then knowing the distance from the center to the midpoint of a side is sufficient, but if it is not, then we lack any information about the length of the other two sides. INSUFFICIENT.

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem.
Remember equal number of variables and independent equations ensures a solution.

The perimeter of square ABCD is half of the perimeter of parallelogram EFGH. What is the perimeter of quadrilateral EFGH?

(1) a line segment drawn from the center of ABCD to the midpoint of BC is 2 inches long

(2) a line segment drawn from the center of EFGH to the midpoint of FG is 4 inches long

Transforming the original condition, square's perimeter=s, parellelgram's perimeter=p, then we have s=p/2 and thus there are 2 variable (s,p) and 1 equation (s=p/2). Thus we need 1 more equation to match the number of variables and equations and D has high probability of being the answer.

In case of 1), s=16 gives us 16=p/2 then p=32. The condition is sufficient
In case of 2), parallelogram itself has 3 variables and the condition is not sufficient. Therefore the answer is A.