D is a point on the side BC of a `DeltaABC` such that AD bisects `angle BAC` . Then

Options

A. `BD=CD`
B. `BA gt BD`
C. `BD gt BA`
D. `CD gt CA`

Answer

To solve the problem, we need to analyze the triangle \( \Delta ABC \) where \( D \) is a point on side \( BC \) such that \( AD \) bisects \( \angle BAC \). We will use the Angle Bisector Theorem to find the relationship between the segments \( BD \) and \( DC \).

### Step 1: Understand the Angle Bisector Theorem
The Angle Bisector Theorem states that if a point \( D \) lies on the side \( BC \) of triangle \( ABC \) and \( AD \) bisects \( \angle BAC \), then the following relationship holds:
\[ \frac{AB}{AC} = \frac{BD}{DC} \]
This means that the ratio of the lengths of the sides opposite the angles is equal to the ratio of the segments created by the angle bisector on the opposite side.

### Step 2: Set Up the Ratios
Let:
- \( AB = c \)
- \( AC = b \)
- \( BD = x \)
- \( DC = y \)

According to the Angle Bisector Theorem, we can write:
\[ \frac{c}{b} = \frac{x}{y} \]

### Step 3: Cross-Multiply
Cross-multiplying gives us:
\[ c \cdot y = b \cdot x \]

### Step 4: Analyze the Implications
From the equation \( c \cdot y = b \cdot x \), we can infer that if \( c > b \) (i.e., \( AB > AC \)), then \( x > y \) (i.e., \( BD > DC \)). Conversely, if \( c < b \), then \( x < y \).

### Step 5: Conclusion
Thus, we can conclude that the lengths of segments \( BD \) and \( DC \) are proportional to the lengths of sides \( AC \) and \( AB \) respectively. Therefore, if \( AB > AC \), then \( BD > DC \) and vice versa.

### Summary of Findings
- If \( AB > AC \), then \( BD > DC \).
- If \( AB < AC \), then \( BD < DC \).
- If \( AB = AC \), then \( BD = DC \).