Question

question 15
given that a quadrilateral pqrs is a parallelogram, \overline {pq} and \overline {rs} are
opposite sides, \overline {pq}=5x + 6, \overline {rs}=2x + 12, and \overline {qr}=25 which of th
following statements are correct?
more than one answer may be correct.
\square \overline {ps}=25
\square perimeter = 82
\square the diagonals are perpendicular
\square x = 4
\square opposite angles are congruent
\square the parallelogram pqrs is a square
\square none of these answers are correct

Explanation:

Step1: Use the property of parallelogram (opposite sides are equal)

In a parallelogram \(PQRS\), \(PQ = RS\). So, \(5x+6=2x + 12\).

Step2: Solve the equation for \(x\)

Subtract \(2x\) from both sides: \(5x-2x+6=2x-2x + 12\), which gives \(3x+6=12\).
Subtract \(6\) from both sides: \(3x+6 - 6=12 - 6\), so \(3x=6\).
Divide both sides by \(3\): \(x=\frac{6}{3}=2\).

Step3: Calculate the lengths of sides

\(PQ=5x + 6=5\times2+6=16\), \(RS=2x + 12=2\times2+12 = 16\), \(QR = 25\) (since \(QR = 2x+12\) when \(x = 2\) is wrong, actually we should use perimeter formula. Perimeter of parallelogram \(P = 2(PQ+QR)\). \(PQ=16\), \(QR = 25\), \(P=2(16 + 25)=82\).
Opposite angles of a parallelogram are congruent. Diagonals of a parallelogram are not perpendicular (unless it is a rhombus). A parallelogram is not a square (a square is a special parallelogram with all sides equal and all angles \(90^{\circ}\)).

Answer:

\(\text{Perimeter}=82\), \(\text{Opposite angles are congruent}\)