Question

tu || sv. find uv.
(there is a triangle-like figure with points w, s, t, u, v. segment lengths: w to v is 15, w to s is 8, s to t is 16. tu is parallel to sv.)

Explanation:

Step1: Identify Similar Triangles

Since \( TU \parallel SV \), triangles \( \triangle WSV \) and \( \triangle WTU \) are similar by the Basic Proportionality Theorem (Thales' theorem). So, the ratios of corresponding sides are equal: \( \frac{WS}{WT} = \frac{WV}{WU} \).

Step2: Calculate \( WT \)

\( WS = 8 \), \( ST = 16 \), so \( WT = WS + ST = 8 + 16 = 24 \). \( WV = 15 \), let \( WU = 15 + UV \), but we can use the ratio \( \frac{8}{24} = \frac{15}{WU} \). Wait, no, correct ratio: \( \frac{WS}{WT} = \frac{WV}{WU} \) → \( \frac{8}{24} = \frac{15}{WU} \)? Wait, no, \( WV \) is 15, \( WU \) is \( WV + VU \)? Wait, no, looking at the diagram: \( WV = 15 \), \( WU \) is the side from \( W \) to \( U \), and \( WS = 8 \), \( ST = 16 \). So similar triangles: \( \triangle WSV \sim \triangle WTU \), so \( \frac{WS}{WT} = \frac{WV}{WU} \). \( WT = 8 + 16 = 24 \), \( WS = 8 \), \( WV = 15 \). So \( \frac{8}{24} = \frac{15}{WU} \)? Wait, no, that would be inverted. Wait, \( SV \parallel TU \), so corresponding sides: \( WS \) corresponds to \( WT \), \( WV \) corresponds to \( WU \). So \( \frac{WS}{WT} = \frac{WV}{WU} \) → \( \frac{8}{24} = \frac{15}{WU} \)? Solving for \( WU \): \( WU = \frac{15 \times 24}{8} = 45 \). Then \( UV = WU - WV = 45 - 15 = 30 \). Wait, let's check again. \( WS = 8 \), \( WT = 8 + 16 = 24 \). The ratio of \( WS \) to \( WT \) is \( 8/24 = 1/3 \). So the ratio of \( WV \) to \( WU \) should also be \( 1/3 \). So \( WV / WU = 1/3 \) → \( 15 / WU = 1/3 \) → \( WU = 45 \). Then \( UV = WU - WV = 45 - 15 = 30 \).

Step3: Solve for \( UV \)

From the ratio \( \frac{WS}{WT} = \frac{WV}{WU} \), \( \frac{8}{24} = \frac{15}{WU} \) → \( WU = \frac{15 \times 24}{8} = 45 \). Then \( UV = WU - WV = 45 - 15 = 30 \).

Answer:

30