In the given figure, $PQ \parallel YZ$ such that $XP : PY = 2 : 3$. If $PQ = 5\text{ cm}$, then $YZ$ equals
Show Hint
A common mistake is using the ratio $\frac{XP}{PY} = \frac{PQ}{YZ}$.
Remember, similarity ratios must compare the sides of the smaller triangle ($\Delta XPQ$) to the entire side of the larger triangle ($\Delta XYZ$), i.e., $XP$ to $XY = XP + PY$.
Step 1: Think in terms of a scale factor rather than solving a proportion equation.
Since $XP:PY = 2:3$, the whole side $XY = XP+PY$ is split as $2:3$, so $XP:XY = 2:5$. Step 2: Identify the scale factor from the small triangle to the big one.
Triangle $XPQ$ is similar to triangle $XYZ$ with scale factor $\frac{XY}{XP} = \frac{5}{2}$ going from the smaller triangle to the larger one. Step 3: Scale PQ up to get YZ. \[ YZ = PQ \times \frac{5}{2} = 5 \times \frac{5}{2} = 12.5\text{ cm} \]
\[ \boxed{YZ = 12.5\text{ cm}} \]
