Question

given that the quadrilateral pqrs is a parallelogram, \\( \overline { s t } = 6 x + 5 \\) and \\( \overline { q t } = 4 x + 31 \\), what is the length of \\( \overline { q s } \\)?

Explanation:

Step1: Use the property of parallelogram diagonals

In a parallelogram, the diagonals bisect each other. So, \(ST = QT\).
Given \(ST=6x + 5\) and \(QT = 4x+31\), we set up the equation \(6x + 5=4x + 31\).

Step2: Solve the equation for \(x\)

Subtract \(4x\) from both sides: \(6x-4x + 5=4x-4x + 31\), which simplifies to \(2x+5 = 31\).
Subtract \(5\) from both sides: \(2x+5 - 5=31 - 5\), so \(2x=26\).
Divide both sides by \(2\): \(x=\frac{26}{2}=13\).

Step3: Find the length of \(QT\)

Substitute \(x = 13\) into the expression for \(QT\): \(QT=4x+31=4\times13 + 31\).
First, calculate \(4\times13=52\), then \(52+31 = 83\).

Step4: Calculate the length of \(QS\)

Since \(QS = 2\times QT\) (because the diagonals bisect each other), \(QS=2\times83 = 166\).

Answer:

\(166\)