P is a point inside the triangle ABC. Lines are drawn through P, parallel to the sides of the triangle.The three resulting triangles with the vertex at P have areas `4,9 and 49 sq`. units. The area of the triangle ABC is -

Options

A. `2sqrt(3)`
B. 12
C. 24
D. 144

Answer

To find the area of triangle ABC given that point P is inside the triangle and the areas of the smaller triangles formed by lines drawn through P parallel to the sides of triangle ABC are 4, 9, and 49 square units, we can follow these steps:

### Step 1: Understand the relationship between the areas
The areas of the triangles formed by point P and the sides of triangle ABC are related to the area of triangle ABC. If we denote the areas of the triangles as follows:
- Area of triangle PAB = 4 sq. units
- Area of triangle PBC = 9 sq. units
- Area of triangle PCA = 49 sq. units

### Step 2: Calculate the area of triangle ABC
The area of triangle ABC can be calculated using the areas of the smaller triangles. The area of triangle ABC is equal to the sum of the areas of the smaller triangles plus the area of the triangle formed by point P and the vertices of triangle ABC.

Using the formula:
\[ \text{Area of } \triangle ABC = \text{Area of } \triangle PAB + \text{Area of } \triangle PBC + \text{Area of } \triangle PCA + \text{Area of triangle at P} \]

However, since the area at P is not directly given, we can use the relationship that the area of triangle ABC is proportional to the areas of the smaller triangles.

### Step 3: Find the ratio of the areas
The areas of the triangles are proportional to the squares of the lengths of the segments from point P to the sides of triangle ABC. Thus, we can express the area of triangle ABC as:
\[ \text{Area of } \triangle ABC = k \cdot (A_1 + A_2 + A_3) \]
where \( A_1, A_2, A_3 \) are the areas of the smaller triangles (4, 9, and 49), and \( k \) is a constant that relates the areas.

### Step 4: Calculate the total area
Now, we can calculate the total area:
\[ \text{Area of } \triangle ABC = 4 + 9 + 49 = 62 \text{ sq. units} \]

### Step 5: Find the area of triangle ABC using the ratios
The area of triangle ABC can also be calculated using the formula:
\[ \text{Area of } \triangle ABC = \sqrt{A_1} + \sqrt{A_2} + \sqrt{A_3} \]
This gives us:
\[ \text{Area of } \triangle ABC = \sqrt{4} + \sqrt{9} + \sqrt{49} = 2 + 3 + 7 = 12 \]
However, this is not the area of triangle ABC. We need to square this result to find the area of triangle ABC:
\[ \text{Area of } \triangle ABC = (2 + 3 + 7)^2 = 12^2 = 144 \text{ sq. units} \]

### Final Answer
Thus, the area of triangle ABC is:
\[ \text{Area of } \triangle ABC = 144 \text{ sq. units} \]