Question

find vw.
vw =
submit
work it out
not feeling

Explanation:

Step1: Identify Similar Triangles

Since \( VS \parallel UT \) (by the midline or proportional segments theorem), triangles \( \triangle WVS \) and \( \triangle WUT \) are similar. So, the ratios of corresponding sides are equal. Let \( VW = x \), then \( WU = VW + VU = x + 32 \). The ratio of \( WS \) to \( WT \) is \( \frac{WS}{WT}=\frac{30 - 20}{30}=\frac{10}{30}=\frac{1}{3} \)? Wait, no, wait. Wait, the segments: \( WT \) is 30? Wait, no, the diagram: \( WT \) has a segment of 20 and the total is 30? Wait, no, maybe \( WT = 30 \), \( ST = 20 \), so \( WS = WT - ST = 30 - 20 = 10 \)? Wait, no, maybe the other way. Wait, the sides: \( VU = 32 \), \( UT \) is a base, \( VS \) is parallel to \( UT \), so by the Basic Proportionality Theorem (Thales' theorem), \( \frac{WV}{WU}=\frac{WS}{WT} \). Wait, let's re-express. Let \( WV = x \), so \( WU = WV + VU = x + 32 \). The sides \( WS \) and \( WT \): \( WT = 30 \), \( ST = 20 \), so \( WS = WT - ST = 30 - 20 = 10 \)? No, that can't be. Wait, maybe \( WT = 30 \), \( WS = 20 \)? Wait, the diagram: the right side has a segment of 20 (from S to T) and the total from W to T is 30? No, maybe the labels: \( W \) at top, \( T \) at bottom right, \( S \) on \( WT \), \( V \) on \( WU \), \( U \) at bottom left. So \( WU = 32 + x \) (where \( x = WV \)), \( WT = 30 \), \( WS = 20 \), \( ST = 10 \)? No, that doesn't make sense. Wait, maybe the ratio is \( \frac{WV}{WU}=\frac{WS}{WT} \). Wait, if \( VS \parallel UT \), then \( \triangle WVS \sim \triangle WUT \), so \( \frac{WV}{WU}=\frac{WS}{WT} \). Let's assume \( WS = 20 \), \( WT = 30 \), so \( \frac{WV}{WV + 32}=\frac{20}{30}=\frac{2}{3} \). Then solve for \( WV \):

Step2: Set Up Proportion

Let \( WV = x \). Then \( WU = x + 32 \). Since \( \triangle WVS \sim \triangle WUT \), \( \frac{x}{x + 32}=\frac{20}{30}=\frac{2}{3} \). Cross-multiplying: \( 3x = 2(x + 32) \)

Step3: Solve for \( x \)

\( 3x = 2x + 64 \)

Subtract \( 2x \) from both sides: \( x = 64 \)? Wait, no, that can't be. Wait, maybe I got the ratio reversed. Maybe \( \frac{WV}{VU}=\frac{WS}{ST} \). Wait, no, 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. So \( \frac{WV}{VU}=\frac{WS}{ST} \). Let's check: \( VU = 32 \), \( ST = 30 - 20 = 10 \)? No, the diagram has a segment of 30 (the total) and 20 (from S to T), so \( WS = 30 - 20 = 10 \), \( ST = 20 \)? No, that's 10 and 20, sum 30. Then \( \frac{WV}{32}=\frac{10}{20}=\frac{1}{2} \), so \( WV = 16 \). Wait, that makes more sense. Let's re-express:

By Thales' theorem (Basic Proportionality Theorem), since \( VS \parallel UT \), \( \frac{WV}{VU}=\frac{WS}{ST} \). Here, \( WS = 30 - 20 = 10 \), \( ST = 20 \), \( VU = 32 \). So \( \frac{WV}{32}=\frac{10}{20}=\frac{1}{2} \). Therefore, \( WV = 32 \times \frac{1}{2} = 16 \). Wait, but then \( WU = WV + VU = 16 + 32 = 48 \), and \( \frac{WS}{WT}=\frac{10}{30}=\frac{1}{3} \), \( \frac{WV}{WU}=\frac{16}{48}=\frac{1}{3} \), which matches. So that's correct.

Step1: Apply Thales' Theorem

Since \( VS \parallel UT \), \( \frac{WV}{VU}=\frac{WS}{ST} \). \( WS = 30 - 20 = 10 \), \( ST = 20 \), \( VU = 32 \).

Step2: Solve for \( WV \)

\( \frac{WV}{32}=\frac{10}{20} \)

\( WV = 32 \times \frac{10}{20} = 32 \times \frac{1}{2} = 16 \)

Wait, but the question is \( VW \), which is \( WV \), so \( VW = 16 \)? Wait, no, maybe I mixed up \( WV \) and \( VW \). \( VW \) is \( WV \), so yes. Wait, but let's check again. If \( VW = 16 \), then \( WU = 16 + 32 = 48 \), and \( \frac{VW}{WU}=\frac{16}{48}=\frac{1}{3} \), and \( \frac{WS}{WT}=\frac{10}{30}=\frac{1}{3} \), so the ratios are equal, so the triangles are similar. So that works.

Wait, maybe the initial mistake was in the segment labels. Let's confirm: the side from W to T is 30 units, with S dividing it into WS = 10 and ST = 20? No, that would be WS = 10, ST = 20, total 30. Then the side from W to U is 32 + VW, and since VS is parallel to UT, the ratio of VW to WU is equal to the ratio of WS to WT, which is 10/30 = 1/3. So \( \frac{VW}{VW + 32}=\frac{1}{3} \)

Multiply both sides by \( 3(VW + 32) \):

\( 3VW = VW + 32 \)

Subtract \( VW \):

\( 2VW = 32 \)

\( VW = 16 \)

Yes, that's correct. So the key was recognizing the similar triangles and applying the proportionality.

Answer:

\( \boxed{16} \)