Question

given that a quadrilateral pqrs is a parallelogram, \overline{pq} and \overline{rs} are opposite sides, \overline{pq}=5x + 11, \overline{rs}=2x + 20, and \overline{qr}=33 which of t more than one answer may be correct. \square the parallelogram pqrs is a rhombus \square opposite angles are congruent \square \overline{qr}=33 \square perimeter = 118 \square x = 4 \square the diagonals are perpendicular \square none of these answers are correct

Explanation:

Step1: Use the property of parallelogram

In a parallelogram, opposite sides are equal. So, \(PQ = RS\) and \(QR=PS\). Given \(PQ = 5x + 11\) and \(RS=2x + 20\).
Set up the equation \(5x+11 = 2x + 20\).

Step2: Solve the equation for \(x\)

Subtract \(2x\) from both sides: \(5x-2x+11=2x - 2x+20\), which gives \(3x+11 = 20\).
Subtract \(11\) from both sides: \(3x+11 - 11=20 - 11\), so \(3x=9\).
Divide both sides by \(3\): \(x=\frac{9}{3}=3\).

Step3: Find the length of \(QR\)

If \(x = 3\), then \(QR=33\) (since \(QR\) is given as \(33\) in the problem statement and we just verified \(x\) value).

Step4: Check if it is a rhombus

For a parallelogram to be a rhombus, all sides must be equal. \(PQ=5x + 11=5\times3+11=15 + 11=26\), \(QR = 33\). Since \(26
eq33\), it is not a rhombus.

Step5: Check opposite angles property

The problem does not give any information about angles, so we cannot confirm the opposite - angles - are - congruent property from the given side - length equations.

Step6: Calculate the perimeter

If \(PQ = 26\) and \(QR = 33\), perimeter of parallelogram \(P = 2(PQ + QR)=2(26 + 33)=2\times59 = 118\).

Answer:

\(\boxed{QR = 33}\), \(\boxed{\text{Perimeter}=118}\)