Question

given a quadrilateral pqrs is a parallelogram, pq and rs are opposite sides, pq = 6x + 15, rs = 3x + 30, and qr = 40 which of t

Explanation:

Step1: Use the property of opposite sides of a parallelogram

In a parallelogram, opposite sides are equal. So, \(PQ = RS\).
Given \(PQ=6x + 15\) and \(RS = 3x+30\).
Set up the equation \(6x + 15=3x + 30\).
Subtract \(3x\) from both sides: \(6x-3x+15=3x - 3x+30\), which gives \(3x+15 = 30\).
Subtract \(15\) from both sides: \(3x+15 - 15=30 - 15\), so \(3x=15\).
Divide both sides by \(3\): \(x=\frac{15}{3}=5\).

Step2: Calculate the length of \(PQ\)

Substitute \(x = 5\) into \(PQ=6x + 15\).
\(PQ=6\times5+15=30 + 15=45\).

Step3: Calculate the perimeter

Since \(PQ = RS = 45\) and \(QR=40\) (because \(QR\) is adjacent to \(PQ\) and in a parallelogram \(PQ\parallel RS\), \(QR\parallel PS\) and \(QR = PS\)).
The perimeter \(P=2(PQ + QR)\).
\(P=2(45 + 40)=2\times85 = 170\).

Step4: Analyze the properties of parallelogram and rhombus

• A parallelogram: has opposite sides equal and parallel. PQRS is a parallelogram (given).
• A rhombus: is a parallelogram with all sides equal. Here \(PQ = 45\), \(QR=40\), so \(PQ

eq QR\), so PQRS is not a rhombus.

• Diagonals of a parallelogram: bisect each other. Diagonals of a rhombus (a special - case of parallelogram) are perpendicular. Since PQRS is not a rhombus, we cannot say diagonals are perpendicular.

Answer:

• Opposite sides are congruent
• Perimeter \(=170\)
• \(\overline{PQ}=45\)