If the sides of triangle ABC are such that `a=4 b=5,c=6,` then the ratio in which incentre divide the angle bisector of B is

Options

A. `2:3`
B. `2:1`
C. `5:2`
D. `1:1`

Answer

To find the ratio in which the incenter divides the angle bisector of angle B in triangle ABC, where the sides are given as \( a = 4 \), \( b = 5 \), and \( c = 6 \), we can follow these steps:

### Step 1: Understand the triangle and its sides
We have triangle ABC with:
- Side \( a \) opposite angle \( A \) = 4 (BC)
- Side \( b \) opposite angle \( B \) = 5 (AC)
- Side \( c \) opposite angle \( C \) = 6 (AB)

### Step 2: Identify the angle bisector
The angle bisector of angle \( B \) divides the angle into two equal parts. Let’s denote the point where the angle bisector intersects side \( AC \) as point \( I \) (the incenter).

### Step 3: Use the property of the incenter
The property of the incenter states that the ratio in which the incenter divides the angle bisector can be expressed as:
\[ \frac{BI}{IC} = \frac{a + c}{b} \]
where \( a \) and \( c \) are the lengths of the sides adjacent to angle \( B \), and \( b \) is the length of the side opposite angle \( B \).

### Step 4: Substitute the values
From the given values:
- \( a = 4 \)
- \( b = 5 \)
- \( c = 6 \)

Substituting these values into the formula:
\[ \frac{BI}{IC} = \frac{a + c}{b} = \frac{4 + 6}{5} = \frac{10}{5} = 2 \]

### Step 5: Write the final ratio
Thus, the ratio in which the incenter divides the angle bisector \( BI \) is:
\[ \frac{BI}{IC} = 2:1 \]

### Conclusion
The incenter divides the angle bisector of angle \( B \) in the ratio \( 2:1 \).

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