Final answer:
By setting the given expressions for the lengths of opposite sides PQ and RS equal in a parallelogram, we solve for 'x' and find that both sides are 32 units, and thus PS must also be 42 units like QR. PQRS is a parallelogram with opposite sides congruent, but not a rhombus since not all sides are equal.
Step-by-step explanation:
You have presented a problem involving a parallelogram named PQRS with the given side lengths of PQ and RS, in addition to the length of QR. We know that in a parallelogram, opposite sides are congruent, so we can set the lengths of PQ and RS equal to each other to find the value of 'x'.
PQ = RS
6x + 14 = 5x + 17
6x - 5x = 17 - 14
x = 3
Now, we can use the value of 'x' to find the actual lengths of PQ and RS:
PQ = 6(3) + 14 = 18 + 14 = 32
RS = 5(3) + 17 = 15 + 17 = 32
Since PQ and RS are equal and we are given that QR = 42, we know that PS must also equal 42 (statement a is correct) because opposite sides of a parallelogram are congruent (statement b is also correct). As for statement c, we do not have enough information about the diagonals being perpendicular; this is not a general characteristic of parallelograms, so we cannot confirm it. For statement d, a rhombus is a parallelogram with all sides of equal length. Since QR and PS are longer than PQ and RS, PQRS cannot be a rhombus (statement d is incorrect).