E is the mid-point of the median AD of a `DeltaABC`. If B is extended it meets the side AC at F, then CF is equal to

Options

A. `AC//3`
B. 2AC/3`
C. `AC//2`
D. None of these

Answer

To solve the problem, we need to analyze the triangle \( \Delta ABC \) and its median \( AD \). Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Triangle and Median
- We have triangle \( \Delta ABC \).
- \( D \) is the midpoint of side \( BC \), which means \( BD = DC \).
- \( E \) is the midpoint of the median \( AD \).

**Hint:** Remember that the median of a triangle divides the opposite side into two equal segments.

### Step 2: Extend Line \( B \) to Meet \( AC \) at Point \( F \)
- Extend line \( B \) such that it intersects side \( AC \) at point \( F \).

**Hint:** Visualize the triangle and the extended line to understand the relationships between the points.

### Step 3: Draw Parallel Lines
- Draw a line from point \( D \) to point \( G \) on line \( AC \) such that \( DG \) is parallel to \( BF \).
- Since \( E \) is the midpoint of \( AD \), \( DE \) will also be parallel to \( BF \).

**Hint:** Use the properties of parallel lines to establish relationships between segments.

### Step 4: Establish Relationships in Triangles
- In triangle \( ADG \), since \( E \) is the midpoint of \( AD \), we have \( AE = ED \).
- Since \( DG \) is parallel to \( BF \), triangles \( AEF \) and \( DGF \) are similar.

**Hint:** Similar triangles have proportional sides. Use this property to find relationships between segments.

### Step 5: Analyze the Proportions
- Since \( BD = DC \) (as \( D \) is the midpoint), we can say that \( FG \) is equal to \( GC \).
- Therefore, we can conclude that \( FG = GC \).

**Hint:** Midpoints and equal segments help in establishing ratios.

### Step 6: Divide Segment \( AC \)
- The segment \( AC \) is divided into three equal parts: \( AF \), \( FG \), and \( GC \).
- Thus, if we denote each part as \( x \), we have \( AF = FG = GC = x \).

**Hint:** Recognize that if \( AC \) is divided into three equal parts, then the length of \( CF \) will be two parts of \( AC \).

### Step 7: Calculate \( CF \)
- Since \( CF \) consists of two segments \( FG \) and \( GC \), we have \( CF = FG + GC = x + x = 2x \).
- Since \( AC = AF + FG + GC = x + x + x = 3x \), we can express \( CF \) in terms of \( AC \):
\[ CF = \frac{2}{3} AC \]

**Hint:** Use the total length of \( AC \) to express \( CF \) in a fractional form.

### Conclusion
Thus, we find that \( CF \) is equal to \( \frac{2}{3} AC \).

**Final Answer:** \( CF = \frac{2}{3} AC \)