If AD is a median of a `Delta`ABC and P is a point on AC such that ar `(DeltaADP)` : ar `(Delta ABD)` = 2 : 3, then ar `(Delta PDC)` : ar `(Delta ABC)` is

Options

A. `1:6`
B. `1:5`
C. `2:5`
D. `3:5`

Answer

To solve the problem step by step, we will follow the reasoning laid out in the video transcript.

### Step 1: Understand the given information
We are given that AD is a median of triangle ABC, which means it divides triangle ABC into two equal areas, specifically triangles ABD and ADC.

### Step 2: Assign area values
Let the area of triangle ABD be \( x \). Since AD is a median, the area of triangle ADC will also be \( x \). Therefore, the total area of triangle ABC is:
\[ \text{Area of } \triangle ABC = \text{Area of } \triangle ABD + \text{Area of } \triangle ADC = x + x = 2x \]

### Step 3: Use the given ratio of areas
We are given that the ratio of the area of triangle ADP to the area of triangle ABD is \( 2:3 \). This can be expressed mathematically as:
\[ \frac{\text{Area of } \triangle ADP}{\text{Area of } \triangle ABD} = \frac{2}{3} \]
Substituting the area of triangle ABD:
\[ \frac{\text{Area of } \triangle ADP}{x} = \frac{2}{3} \]
From this, we can find the area of triangle ADP:
\[ \text{Area of } \triangle ADP = \frac{2}{3}x \]

### Step 4: Find the area of triangle PDC
Now, we need to find the area of triangle PDC. We know that:
\[ \text{Area of } \triangle PDC = \text{Area of } \triangle ADC - \text{Area of } \triangle ADP \]
Substituting the known areas:
\[ \text{Area of } \triangle PDC = x - \frac{2}{3}x \]
Calculating this gives:
\[ \text{Area of } \triangle PDC = \frac{3}{3}x - \frac{2}{3}x = \frac{1}{3}x \]

### Step 5: Find the ratio of areas
Now we need to find the ratio of the area of triangle PDC to the area of triangle ABC:
\[ \frac{\text{Area of } \triangle PDC}{\text{Area of } \triangle ABC} = \frac{\frac{1}{3}x}{2x} \]
Simplifying this ratio:
\[ = \frac{1}{3} \cdot \frac{1}{2} = \frac{1}{6} \]

### Step 6: Express the ratio in the required form
Thus, the ratio of the area of triangle PDC to the area of triangle ABC is:
\[ \text{Area of } \triangle PDC : \text{Area of } \triangle ABC = 1 : 6 \]

### Final Answer
The final answer is:
\[ \text{ar } \triangle PDC : \text{ar } \triangle ABC = 1 : 6 \]