Question

determining a segment length
segment tq is 26 units long. what is the length of ( overline{qv} )?
26 units
31 units
32 units
8 units

Explanation:

Step1: Find the value of \(x\)

Since \(ST = SQ\) (they are congruent segments in the figure), we have the equation \(3x + 2=4x - 1\).
Subtract \(3x\) from both sides: \(2=x - 1\).
Add \(1\) to both sides: \(x=3\).

Step2: Analyze the relationship between \(TQ\) and \(QV\)

The line \(m\) is the perpendicular bisector of \(SQ\) (by the property of the figure). So, \(\triangle STR\cong\triangle QTR\) and \(\triangle SRV\cong\triangle QRV\). Also, \(TQ = QV\) (by the property of perpendicular bisector in a kite - like figure).
But if we consider another approach: First, we know \(TQ = 26\) units. Since the line \(m\) is the perpendicular bisector of \(SQ\), triangles \(STR\) and \(QTR\) are congruent (by \(HL\) - Hypotenuse - Leg congruence if we consider right - triangles \(STR\) and \(QTR\) with \(SR = QR\) (from \(3x+2\) and \(4x - 1\) when \(x = 3\), \(SR=QR = 11\) and \(TR\) is common). And also, triangles \(SRV\) and \(QRV\) are congruent. So, \(TQ=QV\) (corresponding parts of congruent triangles).

Answer:

\(32\) units