Let `vec(A)D` be the angle bisector of `angle A" of " Delta ABC ` such that `vec(A)D=alpha vec(A)B+beta vec(A)C,` then

Options

A. `alpha = (|vec(AB)|)/(|vec(AB)| + |vec(AC)|), beta = (|vec(AC)|)/(|vec(AB)| + |vec(AC)|)`
B. `alpha =(|vec(AB)| + |vec(AC)|)/(|vec(AB)|), beta =(|vec(AB) + vec(AC)|)/(|vec(AC)|)`
C. `alpha =(|vec(AC)|)/(|vec(AB)|+| vec(AC)|), beta =(|vec(AB)|)/(|vec(AB)| + |vec(AC)|)`
D. `alpha = |(vec(AB))/(vec(AC))|, beta =|(vec(AB))/(vec(AB))|`

Answer

To solve the problem, we need to find the values of \(\alpha\) and \(\beta\) in the expression for the angle bisector \(\vec{AD}\) of triangle \(ABC\). The angle bisector divides the opposite side \(BC\) in the ratio of the lengths of the other two sides \(AB\) and \(AC\).

### Step-by-Step Solution:

1. **Understanding the Angle Bisector**:
The angle bisector \(\vec{AD}\) divides the opposite side \(BC\) in the ratio of the lengths of the sides \(AB\) and \(AC\). This means that if \(D\) is the point on \(BC\) where the angle bisector meets, then:
\[ \frac{BD}{DC} = \frac{AB}{AC} \]

2. **Expressing \(\vec{AD}\)**:
According to the problem, we can express \(\vec{AD}\) as:
\[ \vec{AD} = \alpha \vec{AB} + \beta \vec{AC} \]

3. **Using the Ratio Formula**:
The angle bisector theorem states that the length of the segments created by the angle bisector can be expressed in terms of the lengths of the sides:
\[ \vec{AD} = \frac{|\vec{AC}|}{|\vec{AB}| + |\vec{AC}|} \vec{AB} + \frac{|\vec{AB}|}{|\vec{AB}| + |\vec{AC}|} \vec{AC} \]
Here, we identify:
\[ \alpha = \frac{|\vec{AC}|}{|\vec{AB}| + |\vec{AC}|} \]
\[ \beta = \frac{|\vec{AB}|}{|\vec{AB}| + |\vec{AC}|} \]

4. **Final Expression**:
Thus, we have derived the values for \(\alpha\) and \(\beta\):
\[ \alpha = \frac{|\vec{AC}|}{|\vec{AB}| + |\vec{AC}|} \]
\[ \beta = \frac{|\vec{AB}|}{|\vec{AB}| + |\vec{AC}|} \]

5. **Conclusion**:
From the derived expressions, we can conclude that the values of \(\alpha\) and \(\beta\) correspond to the coefficients of \(\vec{AB}\) and \(\vec{AC}\) in the expression for the angle bisector \(\vec{AD}\).