Question

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

Explanation:

Step1: Recall perimeter formula for parallelogram

In a parallelogram, opposite - sides are equal. So the perimeter $P$ of parallelogram $PQSR$ is $P = 2(PQ)+2(RS)$. Given $PQ=(4x - 1)$ cm and $RS=(3x + 7)$ cm, and $P = 74$ cm. Then $74=2(4x - 1)+2(3x + 7)$.

Step2: Expand the equation

Expand the right - hand side: $74 = 8x-2 + 6x + 14$.

Step3: Combine like terms

Combine the $x$ terms and the constant terms: $74=(8x + 6x)+(-2 + 14)$, so $74 = 14x+12$.

Step4: Solve for $x$

Subtract 12 from both sides: $74−12 = 14x$, i.e., $62 = 14x$. Then $x=\frac{62}{14}=\frac{31}{7}$.

Step5: Find the length of a side

Let's assume we want to find the length of $QS$. First, we need to find the lengths of the sides using $x$. Let's find the length of $PQ = 4x - 1=4\times\frac{31}{7}-1=\frac{124}{7}-1=\frac{124 - 7}{7}=\frac{117}{7}$ and $RS = 3x + 7=3\times\frac{31}{7}+7=\frac{93}{7}+7=\frac{93 + 49}{7}=\frac{142}{7}$. But we made a wrong start above. Since opposite sides of a parallelogram are equal, $PQ = RS$. So $4x-1 = 3x + 7$.

Step6: Solve the correct equation for $x$

Subtract $3x$ from both sides: $4x-3x-1=3x-3x + 7$, we get $x=8$.

Step7: Find the length of a side

If $x = 8$, then $PQ=4x - 1=4\times8-1=31$ cm and $RS = 3x + 7=3\times8 + 7=31$ cm. Let the other pair of sides have length $y$. The perimeter $P = 2(PQ)+2y$. So $74=2\times31+2y$.

Step8: Solve for $y$

$74 = 62+2y$. Subtract 62 from both sides: $74 - 62=2y$, i.e., $12 = 2y$. Then $y = 6$ cm. Since $QS$ is one of the non - equal sides (assuming $PQ$ and $RS$ are the longer pair of opposite sides), $QS = 6$ cm.

Answer:

B. 6 cm