Question

in the figure, pq = qr = rs. find tu. the length of tu is

Explanation:

Step1: Apply the theorem of parallel lines cutting off equal line - segments

Since \(PQ = QR=RS\) and the lines are parallel (\(SP\parallel RZ\parallel SY\parallel UW\)), by the theorem of parallel lines cutting off equal line - segments, we have \(XY = YZ = ZS\).

Step2: Use the property of similar triangles or the ratio of line - segments

We know that \(\frac{TX}{XY}=\frac{TU}{US}\). Also, from the given \(XY = 1.9\) and \(TX = 3.6\).
Since the lines are parallel and \(PQ = QR = RS\), we can use the fact that if three parallel lines cut off equal segments on one transversal, then they cut off equal segments on any transversal.
Let's consider the transversal \(TW\) and \(PU\).
We have \(\frac{TX}{XY}=\frac{TU}{US}\). But more simply, because of the equal - segment property of parallel lines:
\(TU=3.6 + 1.9+1.9\)

Answer:

\(7.4\)