If AD is the internal angle bisector of `DeltaABC` with AB = 3 cm and AC = 1 cm, then what is BD : BC equla to ?

Options

A. `1 : 3`
B. `1 : 4`
C. `2 : 3`
D. `3 : 4`

Answer

To solve the problem, we will use the Angle Bisector Theorem, which states that the ratio of the lengths of the two segments created by the angle bisector on the opposite side is equal to the ratio of the lengths of the other two sides of the triangle.

**Step-by-step Solution:**

1. **Identify the Given Information**:
- We have triangle ABC with sides AB = 3 cm and AC = 1 cm.
- AD is the internal angle bisector of angle A.

2. **Apply the Angle Bisector Theorem**:
According to the Angle Bisector Theorem:
\[ \frac{AB}{AC} = \frac{BD}{DC} \]
Here, AB = 3 cm and AC = 1 cm.

3. **Set Up the Ratio**:
Substitute the values into the equation:
\[ \frac{3}{1} = \frac{BD}{DC} \]
This implies:
\[ BD = 3 \cdot DC \]

4. **Express BC in Terms of BD and DC**:
The total length of BC can be expressed as:
\[ BC = BD + DC \]

5. **Substitute BD in Terms of DC**:
From the previous step, we know:
\[ BD = 3 \cdot DC \]
Therefore, substituting this into the equation for BC:
\[ BC = 3 \cdot DC + DC = 4 \cdot DC \]

6. **Find the Ratio of BD to BC**:
Now we can find the ratio of BD to BC:
\[ \frac{BD}{BC} = \frac{3 \cdot DC}{4 \cdot DC} \]
The DC cancels out:
\[ \frac{BD}{BC} = \frac{3}{4} \]

7. **Final Answer**:
Therefore, the ratio \( BD : BC \) is:
\[ 3 : 4 \]