Question

the perimeter of parallelogram pqsr is 74 cm. what is qs?
4 cm
6 cm
8 cm
12 cm

Explanation:

Step1: Recall parallelogram side - length property

In a parallelogram, opposite sides are equal. So \(PQ = SR\) and \(PS=QR\). The perimeter \(P\) of a parallelogram is \(P = 2(PQ + QR)\). Given \(PQ=(4x - 1)\text{ cm}\) and \(SR=(3x + 7)\text{ cm}\), and \(P = 74\text{ cm}\). Since \(PQ=SR\), we first find \(x\) from the perimeter formula.

Step2: Set up the perimeter equation

\(P=2((4x - 1)+(3x + 7))\). Substitute \(P = 74\) into the equation: \(74=2((4x - 1)+(3x + 7))\). First, simplify the expression inside the parentheses: \((4x - 1)+(3x + 7)=4x-1 + 3x+7=7x + 6\). So the equation becomes \(74 = 2(7x + 6)\).

Step3: Solve the equation for \(x\)

Divide both sides of \(74 = 2(7x + 6)\) by 2: \(\frac{74}{2}=7x + 6\), which gives \(37=7x + 6\). Subtract 6 from both sides: \(37-6 = 7x\), so \(31 = 7x\), and \(x=\frac{31}{7}\). But we made a wrong - start. Since \(PQ = SR\) (opposite sides of parallelogram), we have \(4x-1=3x + 7\).

Step4: Solve for \(x\) using opposite - side equality

Subtract \(3x\) from both sides of \(4x-1=3x + 7\): \(4x-3x-1=3x-3x + 7\), which gives \(x=8\).

Step5: Find the length of \(PQ\) and \(QR\)

\(PQ=4x - 1\), substituting \(x = 8\), we get \(PQ=4\times8-1=32 - 1=31\text{ cm}\). \(SR = 3x+7=3\times8 + 7=24 + 7=31\text{ cm}\). Let \(PS = QR = y\). Using the perimeter formula \(P=2(PQ + PS)\), \(74=2(31 + y)\). Divide both sides by 2: \(37=31 + y\). Subtract 31 from both sides: \(y = 6\text{ cm}\).

Answer:

B. 6 cm