Question

find vw.
vw =
submit

Explanation:

Step1: Apply Triangle Proportionality Theorem

The Triangle Proportionality Theorem states that if a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally. Let \( VW = x \). Then the length of the entire side \( WZ \) is \( x + 16 \). The segments on the other side are \( YX = 14 \) and \( ZY = 21 - 14 = 7 \)? Wait, no, looking at the diagram, the vertical side has length 21, with \( YX = 14 \) and \( ZY = 21 - 14 = 7 \)? Wait, no, actually, the vertical side is \( ZX = 21 \), and \( YX = 14 \), so \( ZY = 21 - 14 = 7 \)? Wait, no, maybe I misread. Wait, the line \( VY \) is parallel to \( WX \), so by the Triangle Proportionality Theorem, \( \frac{ZV}{VW} = \frac{ZY}{YX} \). Wait, \( ZV = 16 \), \( VW = x \), \( ZY = 21 - 14 = 7 \)? No, wait, the vertical side: \( ZX = 21 \), \( YX = 14 \), so \( ZY = 21 - 14 = 7 \)? Wait, no, maybe the segments are \( ZY = 14 \) and \( YX = 21 - 14 = 7 \)? Wait, no, the diagram shows \( ZY \) with length 14? Wait, no, the vertical side is 21, with \( YX = 14 \), so \( ZY = 21 - 14 = 7 \)? Wait, I think I got it wrong. Let's re-express. Let’s denote: in triangle \( WZX \), line \( VY \) is parallel to \( WX \). So \( \frac{ZV}{VW} = \frac{ZY}{YX} \). Wait, \( ZV = 16 \), \( VW = x \), \( ZY = 21 - 14 = 7 \)? No, that can't be. Wait, maybe the vertical segments are \( ZY = 14 \) and \( YX = 21 - 14 = 7 \)? No, the problem is that the length from \( Z \) to \( Y \) is 14? Wait, the diagram has \( Z \) at the top, \( X \) at the bottom right, \( W \) at the bottom left. So \( ZX \) is vertical with length 21, \( Y \) is a point on \( ZX \) such that \( YX = 14 \), so \( ZY = 21 - 14 = 7 \). Then \( V \) is on \( ZW \), \( Y \) is on \( ZX \), and \( VY \parallel WX \). So by the Triangle Proportionality Theorem, \( \frac{ZV}{VW} = \frac{ZY}{YX} \). So \( ZV = 16 \), \( VW = x \), \( ZY = 7 \), \( YX = 14 \). So \( \frac{16}{x} = \frac{7}{14} \)? Wait, no, that would be \( \frac{ZV}{ZW} = \frac{ZY}{ZX} \)? Wait, \( ZW = ZV + VW = 16 + x \), \( ZX = 21 \), \( ZY = 21 - 14 = 7 \)? No, I'm confused. Wait, maybe the correct proportion is \( \frac{ZV}{VW} = \frac{ZY}{YX} \), but \( ZY = 14 \) and \( YX = 21 - 14 = 7 \)? No, that doesn't make sense. Wait, let's look again. The vertical side: \( Z \) to \( X \) is 21, \( Y \) is between \( Z \) and \( X \), with \( Y \) to \( X \) being 14, so \( Z \) to \( Y \) is \( 21 - 14 = 7 \). The horizontal side: \( Z \) to \( V \) is 16, \( V \) to \( W \) is \( x \) (what we need to find). Since \( VY \parallel WX \), by the Basic Proportionality Theorem (Thales' theorem), \( \frac{ZV}{VW} = \frac{ZY}{YX} \). So \( \frac{16}{x} = \frac{7}{14} \)? Wait, \( 7/14 = 1/2 \), so \( 16/x = 1/2 \), so \( x = 32 \)? Wait, that can't be. Wait, maybe I mixed up the segments. Let's do it correctly. The Basic Proportionality Theorem: if a line is drawn parallel to one side of a triangle, intersecting the other two sides, then it divides those sides proportionally. So in triangle \( WZX \), line \( VY \parallel WX \), so \( \frac{ZV}{ZW} = \frac{ZY}{ZX} \). Wait, \( ZV = 16 \), \( ZW = 16 + VW \), \( ZY = 21 - 14 = 7 \), \( ZX = 21 \). So \( \frac{16}{16 + VW} = \frac{7}{21} \). Simplify \( \frac{7}{21} = \frac{1}{3} \), so \( \frac{16}{16 + VW} = \frac{1}{3} \). Cross-multiplying: \( 16 \times 3 = 16 + VW \), so \( 48 = 16 + VW \), so \( VW = 48 - 16 = 32 \). Wait, that seems off. Wait, maybe the segments are \( ZY = 14 \) and \( YX = 21 - 14 = 7 \), so \( \frac{ZV}{VW} = \frac{ZY}{YX} \), so \( \frac{16}{VW} = \frac{14}{7} \), which is \( \frac{14}{7} = 2 \), so \( 16 / VW = 2 \), so \( VW = 8 \)? No, that's conflicting. Wait, let's look at the diagram again. The vertical side: \( Z \) at top, \( X \) at bottom right, \( W \) at bottom left. \( ZX = 21 \), \( Y \) is on \( ZX \), \( YX = 14 \), so \( ZY = 21 - 14 = 7 \). \( V \) is on \( ZW \), \( ZV = 16 \), \( VW = x \). Line \( VY \parallel WX \). So by Thales' theorem, \( \frac{ZV}{ZW} = \frac{ZY}{ZX} \). \( ZW = 16 + x \), \( ZY = 7 \), \( ZX = 21 \). So \( \frac{16}{16 + x} = \frac{7}{21} \). Simplify \( \frac{7}{21} = \frac{1}{3} \), so \( 16 / (16 + x) = 1/3 \). Cross multiply: \( 16 \times 3 = 16 + x \), \( 48 = 16 + x \), \( x = 32 \). Wait, but that seems long. Alternatively, maybe the ratio is \( \frac{ZV}{VW} = \frac{ZY}{YX} \), where \( ZY = 14 \) and \( YX = 21 - 14 = 7 \), so \( \frac{16}{x} = \frac{14}{7} = 2 \), so \( x = 8 \). But that contradicts. Wait, maybe I have the segments reversed. Let's consider that \( VY \) is parallel to \( WX \), so triangle \( ZVY \sim \) triangle \( ZWX \) by AA similarity (since \( \angle Z \) is common, and \( \angle ZVY = \angle ZWX \) because \( VY \parallel WX \)). So the ratio of sides should be equal. So \( \frac{ZV}{ZW} = \frac{ZY}{ZX} \). \( ZV = 16 \), \( ZW = 16 + VW \), \( ZY = 21 - 14 = 7 \), \( ZX = 21 \). So \( \frac{16}{16 + VW} = \frac{7}{21} \). \( 7/21 = 1/3 \), so \( 16/(16 + VW) = 1/3 \), so \( 16 + VW = 48 \), so \( VW = 32 \). Alternatively, if \( ZY = 14 \) and \( ZX = 21 \), then \( \frac{ZV}{ZW} = \frac{ZY}{ZX} = \frac{14}{21} = \frac{2}{3} \). So \( \frac{16}{16 + VW} = \frac{2}{3} \). Cross multiply: \( 16 \times 3 = 2(16 + VW) \), \( 48 = 32 + 2VW \), \( 2VW = 16 \), \( VW = 8 \). Ah, here we go! I think I misidentified \( ZY \) and \( YX \). The length from \( Z \) to \( Y \) is 14, and from \( Y \) to \( X \) is \( 21 - 14 = 7 \)? No, wait, the vertical side is \( ZX = 21 \), and \( Y \) is a point on \( ZX \) such that \( ZY = 14 \) and \( YX = 21 - 14 = 7 \)? No, that would mean \( ZY = 14 \) and \( YX = 7 \), but the diagram shows \( YX = 14 \). Wait, the diagram has \( YX = 14 \) (the lower part of the vertical side) and \( ZY = 21 - 14 = 7 \) (the upper part). But that would make the ratio \( ZY/YX = 7/14 = 1/2 \). But if we use similar triangles, triangle \( ZVY \) and triangle \( ZWX \) are similar, so \( ZV/ZW = ZY/ZX \). If \( ZY = 7 \) and \( ZX = 21 \), then \( ZY/ZX = 7/21 = 1/3 \), so \( ZV/ZW = 1/3 \), so \( 16/ZW = 1/3 \), so \( ZW = 48 \), so \( VW = ZW - ZV = 48 - 16 = 32 \). But if \( ZY = 14 \) and \( ZX = 21 \), then \( ZY/ZX = 14/21 = 2/3 \), so \( ZV/ZW = 2/3 \), so \( 16/ZW = 2/3 \), so \( ZW = 24 \), so \( VW = 24 - 16 = 8 \). Now I'm confused. Wait, let's check the problem again. The diagram: \( Z \) at top, \( X \) at bottom right, \( W \) at bottom left. \( ZX \) is vertical with length 21. \( Y \) is on \( ZX \), \( YX = 14 \), so \( ZY = 21 - 14 = 7 \). \( V \) is on \( ZW \), \( ZV = 16 \), \( VW = x \). Line \( VY \parallel WX \). So by Thales' theorem, \( ZV/VW = ZY/YX \). So \( 16/x = 7/14 \). \( 7/14 = 1/2 \), so \( 16/x = 1/2 \), so \( x = 32 \). Yes, that makes sense. Because \( ZY = 7 \), \( YX = 14 \), so the ratio is 7:14 = 1:2. So \( ZV:VW = 1:2 \), so \( 16:VW = 1:2 \), so \( VW = 32 \).

Step2: Solve for \( VW \)

From the Triangle Proportionality Theorem, \( \frac{ZV}{VW} = \frac{ZY}{YX} \). We know \( ZV = 16 \), \( ZY = 21 - 14 = 7 \), \( YX = 14 \). Wait, no, \( ZY = 7 \), \( YX = 14 \), so \( \frac{16}{VW} = \frac{7}{14} \). Simplify \( \frac{7}{14} = \frac{1}{2} \), so \( \frac{16}{VW} = \frac{1}{2} \). Cross-multiplying gives \( VW = 16 \times 2 = 32 \).

Answer:

\( \boxed{32} \)