In the diagram, triangle XYZ is similar to triangle UXS. Given that $US = UX = 9$ cm, $XY = XZ = 20$ cm and $YZ = 12$ cm, find the length of SY. Answer ... cm

1. Identify corresponding sides from the order of similar triangles.

Since $\triangle XYZ \sim \triangle UXS$, the corresponding vertices are
$X \leftrightarrow U,\quad Y \leftrightarrow X,\quad Z \leftrightarrow S.$
So corresponding sides include
$YZ \leftrightarrow XS,$
$XY \leftrightarrow UX,$
$XZ \leftrightarrow US.$

2. Find the length of $XS$ using the similarity scale factor.

Using corresponding sides $XZ \leftrightarrow US$ and $YZ \leftrightarrow XS$,
$\frac{YZ}{XS}=\frac{XZ}{US}.$
Substitute the given lengths:
$\frac{12}{XS}=\frac{20}{9}.$
Hence
$XS=\frac{12\times 9}{20}=5.4\text{ cm}.$

3. Subtract $XS$ from $XY$ to find $SY$.

Since $S$ lies on $XY$,
$SY=XY-XS.$
$SY=20-5.4=14.6\text{ cm}.$