Question

find vw.
t 6 x w
u
vw =

Explanation:

Step1: Identify Similar Triangles

The lines \( TX \) and \( TU \) (with arrows) suggest \( \triangle TXV \sim \triangle TUW \) by the Basic Proportionality Theorem (Thales' theorem), so the ratios of corresponding sides are equal. Let \( VW = x \), then \( UV = 12 - x \). The ratio of \( TX \) to \( TW \) is \( \frac{6}{9}=\frac{2}{3} \), and the ratio of \( UV \) to \( UW \) should be equal (since \( \triangle TXV \sim \triangle TUW \)). Wait, actually, \( TW = 9 \), \( TX = 6 \), \( UW = 12 \). Let \( VW = x \), so \( UV = 12 - x \). By the similarity of triangles, \( \frac{TX}{TW}=\frac{UV}{UW} \)? Wait, no, \( TX \) and \( TW \) are on the same line, \( TX = 6 \), \( XW = 9 - 6 = 3 \)? Wait, no, the top segment is length 9 (from T to W? Wait, the top bar is labeled 9, with T and W? Wait, the diagram: T---X---W, with TX = 6, so XW = 9 - 6 = 3? Wait, no, the top horizontal segment is length 9, with T and W as endpoints, and X is a point on TW with TX = 6. Then the vertical side: U to W is 12, and V is a point on UW, with X connected to V. So \( \triangle TXV \sim \triangle TWU \) (since \( XV \parallel TU \)? Wait, the arrows on \( TX \) and \( TU \) (wait, the arrows are on \( TX \) and \( UV \)? Wait, maybe the triangles are similar by AA similarity. Let's define: Let \( VW = x \), so \( UV = 12 - x \). The ratio of \( TX \) to \( TW \) is \( \frac{6}{9}=\frac{2}{3} \). Since the triangles are similar, the ratio of \( UV \) to \( UW \) should be \( \frac{2}{3} \)? Wait, no, \( TX \) corresponds to \( TW \), and \( UV \) corresponds to \( UW \)? Wait, maybe \( \frac{TX}{TW}=\frac{UV}{UW} \). So \( \frac{6}{9}=\frac{12 - x}{12} \). Solving for \( x \): Cross-multiplying, \( 6 \times 12 = 9 \times (12 - x) \) → \( 72 = 108 - 9x \) → \( 9x = 108 - 72 = 36 \) → \( x = 4 \). Wait, no, that would be \( UV = 12 - x = 8 \), and \( \frac{6}{9}=\frac{8}{12} \) → \( \frac{2}{3}=\frac{2}{3} \), which works. Wait, but we need \( VW \). Wait, maybe I mixed up the sides. Let's re-express: The two triangles are similar, so \( \frac{TX}{TW}=\frac{VW}{UW} \)? No, \( TX = 6 \), \( TW = 9 \), \( UW = 12 \), \( VW = x \). So \( \frac{TX}{TW}=\frac{VW}{UW} \)? Wait, \( \frac{6}{9}=\frac{x}{12} \)? No, that would give \( x = 8 \), but that contradicts. Wait, maybe the correct ratio is \( \frac{XW}{TW}=\frac{VW}{UW} \). \( XW = 9 - 6 = 3 \), so \( \frac{3}{9}=\frac{x}{12} \) → \( x = 4 \). Yes, that makes sense. Because \( XW = 3 \), \( TW = 9 \), so the ratio of \( XW \) to \( TW \) is \( \frac{1}{3} \), and \( VW \) to \( UW \) should be \( \frac{1}{3} \), so \( VW = \frac{1}{3} \times 12 = 4 \). Wait, let's check with similar triangles: \( \triangle XWV \sim \triangle TWU \) (since \( XV \parallel TU \), by the converse of Thales' theorem). So corresponding sides: \( XW \) corresponds to \( TW \), \( VW \) corresponds to \( UW \). So \( \frac{XW}{TW}=\frac{VW}{UW} \). \( XW = 9 - 6 = 3 \), \( TW = 9 \), \( UW = 12 \). So \( \frac{3}{9}=\frac{VW}{12} \) → \( VW = \frac{3 \times 12}{9} = 4 \).

Step2: Solve for \( VW \)

Using the ratio from similar triangles: \( \frac{XW}{TW} = \frac{VW}{UW} \). Substitute \( XW = 9 - 6 = 3 \), \( TW = 9 \), \( UW = 12 \):
\( \frac{3}{9} = \frac{VW}{12} \)
Simplify \( \frac{3}{9} = \frac{1}{3} \), so \( \frac{1}{3} = \frac{VW}{12} \)
Multiply both sides by 12: \( VW = \frac{12}{3} = 4 \)

Answer:

\( 4 \)