If AD is median of triangle ABC and P is a point on AC such that area $( riangle ADP)$ : area $( riangle ABD)$ = 2 : 3, then area $( riangle PDC)$ : area ($ riangle ABC)$ is equal to :

Question

If AD is median of triangle ABC and P is a point on AC such that area $(	riangle ADP)$ : area $(	riangle ABD)$ = 2 : 3, then area $(	riangle PDC)$ : area ($	riangle ABC)$ is equal to :

Accepted Answer

To solve this problem, we need to analyze the given conditions and understand the relationships between the areas of the triangles involved.

### Explanation:

1. **Understanding the Given Information:**
   - AD is the median of triangle ABC. This means that D is the midpoint of BC, and thus, AD divides triangle ABC into two triangles of equal area: $	riangle ABD$ and $	riangle ADC$.
   - We are given that the area of $	riangle ADP$ is to the area of $	riangle ABD$ as 2:3.

2. **Area Relationships:**
   - Since AD is a median, the area of $	riangle ABD$ is equal to the area of $	riangle ADC$. Therefore, the area of $	riangle ABC$ is twice the area of $	riangle ABD$.
   - Let the area of $	riangle ABD$ be 3x. Then, the area of $	riangle ADP$ is 2x (from the given ratio 2:3).
   - Consequently, the area of $	riangle ABC$ is 2 * 3x = 6x.

3. **Finding the Area of $	riangle PDC$:**
   - The area of $	riangle ADC$ is also 3x (since AD is the median).
   - The area of $	riangle ADP$ is 2x, which is part of $	riangle ADC$.
   - Therefore, the area of $	riangle PDC$ is the remaining part of $	riangle ADC$, which is 3x - 2x = x.

4. **Calculating the Required Ratio:**
   - We need to find the ratio of the area of $	riangle PDC$ to the area of $	riangle ABC$.
   - The area of $	riangle PDC$ is x, and the area of $	riangle ABC$ is 6x.
   - Thus, the ratio is x : 6x, which simplifies to 1 : 6.

### Conclusion:

- The correct answer is Option 3, which is 1 : 6. This is because the calculations show that the area of $	riangle PDC$ is one-sixth of the area of $	riangle ABC$.

### Why Other Options are Incorrect:

- **Option 1 (1 : 5):** This would imply that the area of $	riangle PDC$ is one-fifth of the area of $	riangle ABC$, which contradicts our calculation showing it is one-sixth.
- **Option 2 (5 : 1):** This suggests that the area of $	riangle PDC$ is five times the area of $	riangle ABC$, which is not possible given the conditions.
- **Option 4 (6 : 1):** This implies that the area of $	riangle PDC$ is six times the area of $	riangle ABC$, which is also incorrect based on our analysis.

By following the logical steps and calculations, we confirm that Option 3 is indeed the correct answer.