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: Understanding the Question:
We are given a triangle $\Delta XYZ$ with a line segment $PQ$ parallel to the base $YZ$.
The ratio of segments $XP : PY = 2 : 3$.
The length of $PQ$ is given as $5\text{ cm}$. We need to determine the length of the base $YZ$.

Step 2: Key Formula or Approach:
Since $PQ \parallel YZ$, we can apply the AA (Angle-Angle) similarity criterion to triangles $\Delta XPQ$ and $\Delta XYZ$:
- $\angle XPQ = \angle XYZ$ (Corresponding angles)
- $\angle XQP = \angle XZY$ (Corresponding angles)
- $\angle PXQ = \angle YXZ$ (Common angle)
Thus, $\Delta XPQ \sim \Delta XYZ$.
For similar triangles, the ratio of corresponding sides is equal:
\[ \frac{XP}{XY} = \frac{PQ}{YZ} \]

Step 3: Detailed Explanation:

• Express the side $XY$ in terms of the given ratio $XP : PY = 2 : 3$:
Let $XP = 2k$ and $PY = 3k$.
Then, the full length of side $XY$ is:
\[ XY = XP + PY = 2k + 3k = 5k \]

• Determine the ratio of the sides of the similar triangles:
\[ \frac{XP}{XY} = \frac{2k}{5k} = \frac{2}{5} \]

• Set up the proportion with the known length $PQ = 5\text{ cm}$:
\[ \frac{XP}{XY} = \frac{PQ}{YZ} \]
\[ \frac{2}{5} = \frac{5}{YZ} \]

• Solve for $YZ$:
\[ 2 \times YZ = 5 \times 5 \]
\[ 2 \times YZ = 25 \]
\[ YZ = \frac{25}{2} = 12.5\text{ cm} \]


Step 4: Final Answer:
The length of $YZ$ is $12.5\text{ cm}$.