Question

there is a diagram with three vertical lines r, s, t (and their downward counterparts v, x, z) and a transversal line l. the horizontal line m connects v (on r), x (on s), z (on t). the length from v to x is c, x to z is 21. on line l, the segment from u (on r) to w (on s) is 9, and from w (on s) to y (on t) is 27. below the diagram are multiple - choice options: 6, 7, 8, 9.

Explanation:

Step1: Use the property of parallel lines and similar triangles

Since \(r\parallel s\parallel t\), we can use the basic proportionality theorem (Thales' theorem). Let the length of \(VX\) be \(x\). The ratio of the segments on the transversal line \(l\) is equal to the ratio of the segments on the transversal line \(m\). So, \(\frac{UW}{WY}=\frac{VX}{XZ}\).
We know that \(UW = 9\), \(WY=27\), and \(XZ = 21\). Substituting these values into the proportion \(\frac{9}{27}=\frac{x}{21}\).

Step2: Solve the proportion for \(x\)

Cross - multiply the proportion \(\frac{9}{27}=\frac{x}{21}\). We get \(27x=9\times21\). Then \(x=\frac{9\times21}{27}\). Simplify \(\frac{9\times21}{27}=\frac{9\times21}{9\times3}\). Cancel out the common factor of 9, and we have \(x = 7\).

Answer:

7