Question

determining segment lengths
the length of tr is 17 units. what are the lengths of
sv and qt?
sv = units
qt = units

Explanation:

Step1: Find the value of \(x\)

Since \(TR\) is the perpendicular bisector of \(SV\), \(QT = QV\) (by the property of perpendicular bisector in a kite - like figure). So, \(9x - 4=4x + 1\).
Subtract \(4x\) from both sides: \(9x-4x - 4=4x-4x + 1\), which gives \(5x-4 = 1\).
Add \(4\) to both sides: \(5x-4 + 4=1 + 4\), so \(5x=5\).
Divide both sides by \(5\): \(x = 1\).

Step2: Calculate the length of \(SV\)

Since \(TR\) is the perpendicular bisector of \(SV\), \(SR=RV\). And \(SV=2\times RV\). Given \(RV = 3x+2\), substitute \(x = 1\) into \(RV\): \(RV=3\times1 + 2=5\). Then \(SV = 2\times5=10\).

Step3: Calculate the length of \(QT\)

Substitute \(x = 1\) into \(QT=9x - 4\). So \(QT=9\times1-4=5\).

Answer:

\(SV = 10\) units, \(QT = 13\) units.