Question

$\overline{tu} \parallel \overline{sv}$. find $uw$.
(image of a triangle with points u, v, w, s, t; uv = 42, ws = 12, st = 18; $uw = \square$)

Explanation:

Step1: Identify Similar Triangles

Since $\overline{TU} \parallel \overline{SV}$, by the Basic Proportionality Theorem (Thales' theorem), $\triangle WSV \sim \triangle WTU$. So, the ratios of corresponding sides are equal: $\frac{WV}{WU} = \frac{WS}{WT}$.
Let $UW = x$, then $WV = x - 42$. $WS = 12$, $WT = 12 + 18 = 30$.

Step2: Set Up Proportion

Substitute into the proportion: $\frac{x - 42}{x} = \frac{12}{30}$.
Simplify $\frac{12}{30}$ to $\frac{2}{5}$, so $\frac{x - 42}{x} = \frac{2}{5}$.
Cross - multiply: $5(x - 42)=2x$.
Expand: $5x - 210 = 2x$.
Subtract $2x$ from both sides: $3x - 210 = 0$.
Add 210 to both sides: $3x = 210$.
Divide by 3: $x = 70$.

Answer:

70