The median AD of a triangle ABC is bisected at F, and BF is produced to meet the side AC in P. If AP =`lamdaAC`, then what is the value of `lamda`.?

Options

A. `(1)/(4)`
B. `(1)/(2)`
C. `(2)/(3)`
D. `(1)/(3)`

Answer

To solve the problem, we will go through the following steps:

### Step 1: Understand the Triangle and the Median
We have triangle ABC with median AD. The median AD is a line segment from vertex A to the midpoint D of side BC. The median divides the triangle into two smaller triangles of equal area.

### Step 2: Identify the Points
Let F be the midpoint of AD, which means AF = FD. The line BF is extended to meet side AC at point P. We need to find the ratio AP to AC, which is given as λAC.

### Step 3: Set Up the Coordinates
Assume the coordinates of the points:
- A = (h, k)
- B = (0, 0)
- C = (a, 0)

Since D is the midpoint of BC, the coordinates of D will be:
- D = ((0 + a)/2, (0 + 0)/2) = (a/2, 0)

The coordinates of F, being the midpoint of AD, will be:
- F = ((h + a/2)/2, (k + 0)/2) = (h/2 + a/4, k/2)

### Step 4: Determine the Coordinates of Point P
Point P lies on line AC. The equation of line AC can be derived from points A and C. The slope of AC is given by:
- Slope = (0 - k) / (a - h)

Using point-slope form, the equation of line AC is:
- y - k = (0 - k)/(a - h)(x - h)

### Step 5: Find the Intersection Point P
To find point P, we need to express the line BF in terms of its slope and then find where it intersects with line AC. The slope of line BF can be calculated using points B and F.

### Step 6: Set Up the Ratio
Let AP = λAC. We need to express AP in terms of the coordinates we have set up. The length of AC can be calculated using the distance formula:
- AC = √((a - h)² + (0 - k)²)

### Step 7: Solve for λ
Using the coordinates and the lengths calculated, we can set up the equation:
- AP / AC = λ

Substituting the values we derived, we can solve for λ.

### Final Calculation
From the calculations, we find that the ratio AP:AC simplifies to 1:3. Therefore, λ = 1/3.

### Conclusion
Thus, the value of λ is:
- **λ = 1/3**

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