Question

find su.
su =
(there is a triangle with points v, u, s. a line rt is parallel to vu, with rv = 9, vs = 18, and tu = 8.)

Explanation:

Step1: Identify midsegment theorem

Line $RT$ is a midsegment of $\triangle VSU$, since $R$ and $T$ are midpoints (marked by red dots, and $SR=9$, $RV=18$ confirms $R$ divides $SV$ such that $\frac{SR}{SV}=\frac{1}{3}$? No, correction: Wait, the midsegment theorem says if a line connects midpoints of two sides, it's parallel to the third side and half its length. Wait, actually, $\triangle SRT \sim \triangle SVU$ by AA similarity (since $RT \parallel VU$). The ratio of similarity is $\frac{SR}{SV} = \frac{9}{9+18} = \frac{1}{3}$.

Step2: Set up proportion for sides

Since $\triangle SRT \sim \triangle SVU$, $\frac{ST}{SU} = \frac{SR}{SV}$. Let $SU = x$, so $ST = x - 8$.
$\frac{x - 8}{x} = \frac{1}{3}$

Step3: Solve for $x$

Cross-multiply: $3(x - 8) = x$
$3x - 24 = x$
$3x - x = 24$
$2x = 24$
$x = 12$

Answer:

$12$