2021 GMC 12B Problems/Problem 11

Problem

How many of the following statement are true for all parallelogram?

Statement 1: All parallelograms are cyclic quadrilaterals.

Statement 2: All cyclic quadrilaterals are parallelograms.

Statement 3: When all of the midpoint are chosen, the resulting figure is a parallelogram.

Statement 4: The length of a diagonal is the product of two adjacent sides.

Solution

Statement 1: Note that all cyclic parallelograms need opposite supplementary angles. Since this is not the case for all parallelograms, this is false.

Statement 2: This is trivially false.

Statement 3: Since the midpoints of four lines creates four points, the polygon created by these points is a quadrilateral, thus this statement is true.

Statement 4: Note that the length of a diagonal depends on the angles of the parallelagram. Thus, there is no way to find the diagonal without more information than the lengths of the parallelogram, or in other words, this is false.

Since statement 3 is the only correct statement, the answer is .

~pineconee