Question

given that the quadrilateral pqrs is a parallelogram, \\( \overline { s t } = 7 x + 3 \\) a \\( \overline { q t } = 3 x + 43 \\), 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, \(ST = QT\).
\(7x + 3=3x + 43\)

Step2: Solve for \(x\)

Subtract \(3x\) from both sides: \(7x-3x + 3=3x-3x + 43\), \(4x+3 = 43\).
Subtract \(3\) from both sides: \(4x+3 - 3=43 - 3\), \(4x=40\).
Divide both sides by \(4\): \(x=\frac{40}{4}=10\).

Step3: Find the length of \(ST\) or \(QT\)

Substitute \(x = 10\) into \(ST=7x + 3\) (or \(QT = 3x+43\)).
\(ST=7\times10 + 3=70 + 3=73\), \(QT=3\times10+43=30 + 43=73\).

Step4: Calculate the length of \(QS\)

Since \(QS=ST + QT\), \(QS=73+73 = 146\).

Answer:

146