Question

In triangle \( ABC \), the point \( P \) divides \( BC \) internally in the ratio \( 3 : 4 \) and \( Q \) divides \( CA \) internally in the ratio \( 5 : 3 \). If \( AP \) and \( BQ \) intersect in a point \( G \), then \( G \) divides \( AP \) internally in the ratio

• A. \( 2 : 1 \)
• B. \( 5 : 7 \)
• C. \( 7 : 5 \)
• D. \( 1 : 2 \)

Hint:

Van Schooten's or Mass Point Geometry can solve these ratio problems very quickly.

Answer 1

The Correct Option is C

Step 1: Concept Use Ceva's Theorem or Section formula with vector notation for internal division.

Step 2: Meaning Using vector positions: \(p = \frac{4b + 3c}{7}\) and \(q = \frac{3c + 5a}{8}\).

Step 3: Analysis \(G\) lies on \(AP\), so \(g = \frac{m p + 1 a}{m+1}\). \(G\) also lies on \(BQ\), so \(g = \frac{n q + 1 b}{n+1}\).
Substituting \(p\) and \(q\) and equating coefficients of \(\vec{a}, \vec{b}, \vec{c}\).
Solving the ratios gives the position of \(G\) on \(AP\) such that the ratio \(AG:GP = 7:5\).


Step 4: Conclusion The internal ratio is \(7:5\). Final Answer: (C)