Question

find vw.
x
18 22
y w
9
z v
vw =

Explanation:

Step1: Identify the midline theorem

Since \( YW \parallel ZV \) (indicated by the arrows) and \( Y \) divides \( XZ \) such that \( XY = 18 \) and \( YZ = 9 \), so \( \frac{XY}{YZ}=\frac{18}{9} = 2\). By the Midline Theorem (also known as the Basic Proportionality Theorem or Thales' theorem), if a line is drawn parallel to one side of a triangle, intersecting the other two sides, then it divides those sides proportionally. Also, the length of the midline (or the parallel segment) is half the length of the third side if it's a midline, but here we use the proportionality. Let \( VW = x \). The ratio of \( XY \) to \( XZ \) is \( \frac{18}{18 + 9}=\frac{18}{27}=\frac{2}{3}\)? Wait, no, actually, \( Y \) is on \( XZ \) and \( W \) is on \( XV \). Wait, the segments: \( XY = 18 \), \( YZ = 9 \), so \( XZ=XY + YZ=18 + 9 = 27 \). The line \( YW \) is parallel to \( ZV \), so by the Basic Proportionality Theorem, \( \frac{XY}{XZ}=\frac{XW}{XV} \). Wait, \( XV = XW + WV \), and \( XW = 22 \)? Wait, no, the side \( XX \)? Wait, the triangle is \( XZV \)? Wait, the vertices are \( X \), \( Z \), \( V \), with \( Y \) on \( XZ \) and \( W \) on \( XV \), and \( YW \parallel ZV \). So \( XY = 18 \), \( YZ = 9 \), so \( \frac{XY}{YZ}=\frac{18}{9}=2 \), so \( XY:YZ = 2:1 \), so the ratio of similarity is \( \frac{XY}{XZ}=\frac{18}{18 + 9}=\frac{18}{27}=\frac{2}{3} \)? Wait, no, actually, the line \( YW \) is parallel to \( ZV \), so triangle \( XYW \sim \) triangle \( XZV \) by AA similarity (since \( \angle X \) is common and \( \angle XYW=\angle XZV \) because \( YW \parallel ZV \)). So the ratio of corresponding sides is \( \frac{XY}{XZ}=\frac{18}{18 + 9}=\frac{18}{27}=\frac{2}{3} \). Wait, but \( XW \) is 22? Wait, no, the length of \( XW \) is 22? Wait, the diagram shows \( XW = 22 \)? Wait, no, the side from \( X \) to \( W \) is 22? Wait, maybe I misread. Wait, the sides: \( XY = 18 \), \( YZ = 9 \), so \( XZ = 18 + 9 = 27 \). The side \( XW \) is 22? No, wait, the side from \( X \) to \( V \) has segment \( XW = 22 \) and \( WV = x \) (which is \( VW \)). So by the Basic Proportionality Theorem (Thales' theorem), \( \frac{XY}{YZ}=\frac{XW}{WV} \)? Wait, no, Thales' theorem states that if a line is drawn parallel to one side of a triangle, intersecting the other two sides, then it divides those sides proportionally. So \( \frac{XY}{YZ}=\frac{XW}{WV} \). Wait, \( XY = 18 \), \( YZ = 9 \), so \( \frac{18}{9}=\frac{22}{WV} \)? Wait, that would be \( 2=\frac{22}{WV} \), so \( WV = 11 \). Ah, that makes sense. Because \( XY:YZ = 18:9 = 2:1 \), so the ratio of division is 2:1, so \( XW:WV = 2:1 \). Since \( XW = 22 \), then \( WV = \frac{22}{2}=11 \). So \( VW = 11 \).

Step2: Apply the Basic Proportionality Theorem

The Basic Proportionality Theorem (Thales' theorem) states that if a line is drawn parallel to one side of a triangle, intersecting the other two sides, then it divides those sides proportionally. In triangle \( XZV \), line \( YW \) is parallel to \( ZV \), intersecting \( XZ \) at \( Y \) and \( XV \) at \( W \). We have \( XY = 18 \) and \( YZ = 9 \), so the ratio \( \frac{XY}{YZ}=\frac{18}{9}=2 \). This means \( \frac{XW}{WV}=2 \) (since the line \( YW \) divides \( XZ \) and \( XV \) proportionally). Given \( XW = 22 \), we set up the proportion \( \frac{XW}{WV}=2 \), so \( \frac{22}{WV}=2 \). Solving for \( WV \) (which is \( VW \)), we get \( WV=\frac{22}{2}=11 \).

Answer:

\( 11 \)