Question

In the given figure, $PQ \parallel YZ$ such that $XP : PY = 2 : 3$. If $PQ = 5\text{ cm}$, then $YZ$ equals

• A. $12.5\text{ cm}$
• B. $10\text{ cm}$
• C. $15\text{ cm}$
• D. $7.5\text{ cm}$

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 $XY = XP + PY$.

Answer 1

The Correct Option is A

Step 1: Convert the ratio XP:PY into a ratio of XP to the whole side XY.
Since $XP : PY = 2 : 3$, the whole side is divided into $2 + 3 = 5$ equal parts, so $\frac{XP}{XY} = \frac{2}{5}$.
Step 2: Use similarity of triangle XPQ and triangle XYZ.
Because $PQ \parallel YZ$, $\Delta XPQ \sim \Delta XYZ$ by AA similarity, so corresponding sides are in the same ratio: $\frac{PQ}{YZ} = \frac{XP}{XY} = \frac{2}{5}$.
Step 3: Solve for YZ.
Since $PQ = 5\text{ cm}$, we get $\frac{5}{YZ} = \frac{2}{5}$, so $YZ = \frac{5 \times 5}{2} = 12.5\text{ cm}$.
\[ \boxed{YZ = 12.5\text{ cm}} \]