Question

If \(8\overline{AB}-3\overline{AC}-5\overline{AD}=\overline{O}\), then the ratio in which the point \(B\) divides segment \(CD\) is ....

• A. \(8:3\)
• B. \(8:5\)
• C. \(5:3\)
• D. \(3:4\)

Hint:

Take A as origin and write the position vector of B as a weighted average of C and D.

Answer 1

The Correct Option is C

Step 1: Rewrite
$8\vec{b}=3\vec{c}+5\vec{d}$, where the coefficients $3$ and $5$ add up to $8$, so $B$ lies on line $CD$.

Step 2: Use Weights
The point nearer to the larger weight is closer to it, and the weights are $3$ on $C$ and $5$ on $D$, so $CB:BD=5:3$.

Step 3: Match
The ratio $5:3$ matches option (C).

Final Answer:
The ratio is 5:3. \[ \boxed{\text{(C) }5:3} \]