Question

$\overline{st} \parallel \overline{uw}$. find $sw$.
21
t 14 u v
48
s
$sw = \square$
submit

Explanation:

Step1: Identify Similar Triangles

Since \(\overline{ST} \parallel \overline{UW}\), by the Basic Proportionality Theorem (Thales' theorem), the triangles are similar. So, the ratios of corresponding sides are equal. Let \(SW = x\), then \(WV = 48 - x\). The ratio of \(TU\) to \(TV\) should equal the ratio of \(SW\) to \(SV\). Wait, actually, \(TU = 14\), \(TV = 21\), \(SV = 48\), and we need to find \(SW\). So the proportion is \(\frac{TU}{TV}=\frac{SW}{SV}\)? Wait, no, let's check the segments. \(TU = 14\), \(TV = 21\) (since the top segment is 21, from T to V? Wait, the diagram: T to U is 14, U to V is \(21 - 14 = 7\)? Wait, no, the top horizontal segment is 21, with T to U being 14, so U to V is \(21 - 14 = 7\). Then, since \(ST \parallel UW\), triangles \(STV\) and \(UWV\) are similar? Wait, no, maybe \(ST\) and \(UW\) are parallel, so the triangles \(SST\) (wait, no, the vertex is S, so triangle \(STV\) and triangle \(UWV\) are similar? Wait, actually, the sides: \(TU = 14\), \(TV = 21\), so the ratio of \(TU\) to \(TV\) is \(\frac{14}{21}=\frac{2}{3}\). Then, the ratio of \(SW\) to \(SV\) should be the same? Wait, \(SV = 48\), let \(SW = x\), then \(WV = 48 - x\). Wait, maybe the correct proportion is \(\frac{TU}{TV}=\frac{SW}{SV}\)? Wait, no, let's think again. Since \(ST \parallel UW\), the triangles \(SST\) (no, the two triangles: one with base \(ST\) (length 21) and the other with base \(UW\) (length 14)? Wait, no, T to U is 14, T to V is 21, so U to V is 7. Then, the lines \(ST\) and \(UW\) are parallel, so by the Basic Proportionality Theorem, \(\frac{TU}{TV}=\frac{SW}{SV}\)? Wait, no, the segments on the transversal: \(TU = 14\), \(TV = 21\), so the ratio of \(TU\) to \(TV\) is \(\frac{14}{21}=\frac{2}{3}\). Then, the ratio of \(SW\) to \(SV\) should be \(\frac{2}{3}\)? Wait, \(SV = 48\), so \(\frac{SW}{48}=\frac{14}{21}\). Simplify \(\frac{14}{21}=\frac{2}{3}\). So \(\frac{SW}{48}=\frac{2}{3}\). Then, \(SW = 48\times\frac{2}{3}=32\)? Wait, no, wait: if \(TU = 14\), \(TV = 21\), then the ratio of \(TU\) to \(TV\) is \(\frac{14}{21}=\frac{2}{3}\). Then, since \(ST \parallel UW\), the ratio of \(SW\) to \(SV\) is equal to the ratio of \(TU\) to \(TV\)? Wait, \(SV = 48\), so \(SW = 48\times\frac{14}{21}\). Let's calculate that: \(\frac{14}{21}=\frac{2}{3}\), so \(48\times\frac{2}{3}=32\). Wait, but let's check the other way. If \(SW = 32\), then \(WV = 48 - 32 = 16\). Then, the ratio of \(SW\) to \(WV\) is \(32:16 = 2:1\), and the ratio of \(TU\) to \(UV\) is \(14:7 = 2:1\), which matches. So that makes sense. So the proportion is \(\frac{TU}{UV}=\frac{SW}{WV}\), but since \(UV = 21 - 14 = 7\), then \(\frac{14}{7}=\frac{SW}{48 - SW}\), which is \(2=\frac{SW}{48 - SW}\), so \(2(48 - SW)=SW\), \(96 - 2SW = SW\), \(96 = 3SW\), \(SW = 32\). Yes, that works.

Step2: Set Up the Proportion

We have \(TU = 14\), \(UV = 21 - 14 = 7\), and \(SV = 48\). Let \(SW = x\), then \(WV = 48 - x\). Since \(\overline{ST} \parallel \overline{UW}\), the triangles are similar, so \(\frac{TU}{UV}=\frac{SW}{WV}\). Substituting the values: \(\frac{14}{7}=\frac{x}{48 - x}\).

Step3: Solve the Proportion

Simplify \(\frac{14}{7}=2\), so \(2=\frac{x}{48 - x}\). Cross-multiplying gives \(2(48 - x)=x\). Expanding: \(96 - 2x = x\). Adding \(2x\) to both sides: \(96 = 3x\). Dividing both sides by 3: \(x = 32\).

Answer:

32