Question

find vw.
vw = \boxed{}

Explanation:

Step1: Identify Similar Triangles

The lines with red dots are midsegments (or parallel lines), so triangles \( \triangle VYZ \) and \( \triangle WXZ \) are similar by the Basic Proportionality Theorem (Thales' theorem) or AA similarity (since corresponding angles are equal due to parallel lines).

Step2: Set Up Proportion

Let \( VW = x \). Then \( WZ = VW + VZ = x + 16 \). The ratio of corresponding sides in similar triangles is equal. So, \( \frac{VZ}{WZ}=\frac{YZ}{XZ} \). We know \( VZ = 16 \), \( YZ = 14 \), \( XZ = 21 \). Substituting, we get \( \frac{16}{x + 16}=\frac{14}{21} \). Simplify \( \frac{14}{21}=\frac{2}{3} \), so \( \frac{16}{x + 16}=\frac{2}{3} \).

Step3: Solve for \( x \)

Cross - multiply: \( 2(x + 16)=16\times3 \). Expand: \( 2x+32 = 48 \). Subtract 32 from both sides: \( 2x=48 - 32=16 \). Divide by 2: \( x = 8 \)? Wait, no, wait. Wait, maybe I mixed up the sides. Wait, actually, the segment \( YZ = 14 \), \( XZ = 21 \), so the ratio of similarity is \( \frac{YZ}{XZ}=\frac{14}{21}=\frac{2}{3} \). So the ratio of \( VZ \) to \( WZ \) should be \( \frac{2}{3} \). Wait, \( VZ = 16 \), let \( WZ = VW + VZ=VW + 16 \). So \( \frac{16}{VW + 16}=\frac{2}{3} \). Cross - multiply: \( 3\times16 = 2\times(VW + 16) \). \( 48=2VW + 32 \). Subtract 32: \( 16 = 2VW \). Then \( VW = 8 \)? Wait, that seems off. Wait, maybe the ratio is \( \frac{WZ}{VZ}=\frac{XZ}{YZ} \). Let's re - check. If \( \triangle VYZ \sim\triangle WXZ \), then \( \frac{WX}{VY}=\frac{XZ}{YZ}=\frac{WZ}{VZ} \). Wait, maybe I had the ratio reversed. Let's take the larger triangle \( \triangle WXZ \) and smaller \( \triangle VYZ \). So \( \frac{VZ}{WZ}=\frac{YZ}{XZ} \) is incorrect. It should be \( \frac{VZ}{WZ}=\frac{YZ}{XZ} \) only if \( VY\parallel WX \) and \( YZ\parallel XZ \)? No, \( YZ \) and \( XZ \) are the same line? Wait, no, the diagram: \( ZX \) is a vertical segment, \( ZY \) is part of it, \( ZV \) and \( ZW \) are the other sides. Wait, maybe the correct proportion is \( \frac{VZ}{WZ}=\frac{YZ}{XZ} \). Wait, \( YZ = 14 \), \( XZ = 21 \), so \( \frac{14}{21}=\frac{2}{3} \). So \( \frac{VZ}{WZ}=\frac{2}{3} \), \( VZ = 16 \), so \( WZ=\frac{3}{2}\times16 = 24 \). Then \( VW=WZ - VZ=24 - 16 = 8 \)? Wait, that gives \( VW = 8 \)? But let's check again. Wait, maybe the sides: the length from \( Z \) to \( Y \) is 14, \( Z \) to \( X \) is 21, so the ratio of \( ZY \) to \( ZX \) is \( 14/21 = 2/3 \). So the triangle \( VYZ \) is similar to \( WXZ \) with a ratio of \( 2/3 \). So the side \( VZ \) (length 16) corresponds to \( WZ \), so \( VZ/WZ=2/3 \), so \( WZ=(3/2)\times16 = 24 \). Then \( VW=WZ - VZ=24 - 16 = 8 \). Wait, but that seems small. Wait, maybe I made a mistake in identifying the similar triangles. Let's look at the diagram again. The segment with the red dot in the middle of \( VZ \) and \( WX \)? Wait, no, the two red dots: one on \( VW \) (wait, no, the first red dot is on the segment from \( V \) to the middle of \( ZY \), and the second red dot is on \( WX \). Wait, maybe the lines are midsegments, so \( VY \) is parallel to \( WX \) and \( VY=\frac{1}{2}WX \), and \( ZY=\frac{1}{2}ZX \)? Wait, \( ZY = 14 \), \( ZX = 21 \), that's not half. Wait, \( 14/21 = 2/3 \), so maybe the ratio is \( 2/3 \). So if \( ZY = 14 \), \( ZX = 21 \), then the ratio of similarity is \( 2/3 \). So \( VZ/WZ=2/3 \), \( VZ = 16 \), so \( WZ = 24 \), so \( VW=WZ - VZ=24 - 16 = 8 \). Wait, but let's do the proportion again. Let \( VW=x \), so \( WZ=x + 16 \). The ratio of \( ZY \) to \( ZX \) is \( 14/21 = 2/3 \). Since the triangles are similar, the ratio of \( VZ \) to \( WZ \) is also \( 2/3 \). So \( 16/(x + 16)=2/3 \). Cross - multiply: \( 2(x + 16)=48 \), \( 2x+32 = 48 \), \( 2x = 16 \), \( x = 8 \). So \( VW = 8 \)? Wait, but that seems correct? Wait, no, wait, maybe the other way. If the ratio is \( WZ/VZ=3/2 \), then \( WZ=(3/2)\times16 = 24 \), so \( VW=WZ - VZ=24 - 16 = 8 \). Yes, that's correct.

Answer:

\( 8 \)