Question

given that the quadrilateral pqrs is a parallelogram, \\( \overline { s t } = 4 x + 6 \\) and \\( \overline { q t } = 3 x + 12 \\), what is the length of \\( \overline { q s } \\)?
\\( \overline { q s } = \\)______

Explanation:

Step1: Use property of parallelogram diagonals

In a parallelogram, the diagonals bisect each other. So, \(QT = ST\).
Given \(QT=3x + 12\) and \(ST = 4x+6\), we set up the equation \(3x + 12=4x + 6\).

Step2: Solve for \(x\)

Subtract \(3x\) from both sides: \(12=x + 6\).
Subtract \(6\) from both sides: \(x=12 - 6=6\).

Step3: Find length of \(QT\)

Substitute \(x = 6\) into \(QT=3x + 12\). So, \(QT=3\times6+12=18 + 12=30\).

Step4: Find length of \(QS\)

Since \(QS = 2\times QT\) (diagonals bisect each other), \(QS=2\times30 = 60\).

Answer:

\(60\)