Question

In \(\Delta\)ABC, P is a point on AB and Q is a point on AC such that PQ \(\parallel\) BC. If AP : PB = 3 : 2, then PQ : BC is equal to :

• A. 3 : 2
• B. 2 : 5
• C. 3 : 5
• D. 5 : 3

Hint:

Always remember that when a line is drawn parallel to one side of a triangle, the smaller triangle formed is similar to the original larger triangle.
A common mistake is to equate \(\frac{PQ}{BC}\) to \(\frac{AP}{PB}\). This is incorrect!
The parallel segment ratio always corresponds to the ratio of the side of the small triangle to the side of the large triangle, which is:
\[ \frac{\text{Small Side}}{\text{Large Side}} = \frac{AP}{AB} = \frac{AP}{AP + PB} \]

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
In triangle \(ABC\), points \(P\) and \(Q\) lie on sides \(AB\) and \(AC\) respectively, such that \(PQ \parallel BC\). Given the ratio of segments \(AP : PB = 3 : 2\), we need to find the ratio \(PQ : BC\).

Step 2: Key Formula or Approach:
Since \(PQ \parallel BC\), the corresponding angles are equal:
\[ \angle APQ = \angle ABC \quad (\text{Corresponding angles}) \]
\[ \angle AQP = \angle ACB \quad (\text{Corresponding angles}) \]
By AA (Angle-Angle) similarity, \(\Delta APQ \sim \Delta ABC\).
Since corresponding sides of similar triangles are proportional, we have:
\[ \frac{PQ}{BC} = \frac{AP}{AB} \]

Step 3: Detailed Explanation:
1. We are given the ratio:
\[ AP : PB = 3 : 2 \implies \frac{AP}{PB} = \frac{3}{2} \]
Let \(AP = 3x\) and \(PB = 2x\), where \(x\) is a common constant.
2. The total length of side \(AB\) is the sum of segments \(AP\) and \(PB\):
\[ AB = AP + PB = 3x + 2x = 5x \]
3. Since \(\Delta APQ \sim \Delta ABC\), the ratio of the parallel lines \(PQ\) and \(BC\) is equal to the ratio of side \(AP\) to the full side \(AB\):
\[ \frac{PQ}{BC} = \frac{AP}{AB} \]
4. Substitute the values of \(AP\) and \(AB\) into the equation:
\[ \frac{PQ}{BC} = \frac{3x}{5x} = \frac{3}{5} \]
Thus, the ratio \(PQ : BC\) is \(3 : 5\).

Step 4: Final Answer:
The ratio \(PQ : BC\) is equal to \(3 : 5\), which corresponds to option (C).