Assertion (A) : AD is angle bisector of `angleA` of the triangle ABC. If AB = 6 cm, BC = 7 cm, AC = 8 cm then BD = 3 cm and CD = 4 cm.
Reason (R) : The angle bisector AD of the triangle divides base BC in the ratio AB : AC.

Options

A. Both Assertion A and Reason R are correct and Reason R is a correct explanation of Assertion A.
B. Both Assertion A and Reason R are correct but Reason R is not the correct explanation of Assertion A.
C. Assertion A is correct, Reason R is wrong.
D. Assertion A is wrong, Reason R is correct.

Answer

To solve the problem, we will analyze the assertion and reason provided in the question step by step.

### Step 1: Understand the Assertion
The assertion states that AD is the angle bisector of angle A in triangle ABC, with the following dimensions:
- AB = 6 cm
- AC = 8 cm
- BC = 7 cm
- BD = 3 cm
- CD = 4 cm

### Step 2: Verify the Angle Bisector Theorem
According to the Angle Bisector Theorem, the angle bisector of an angle in a triangle divides the opposite side into segments that are proportional to the other two sides. This means:

\[ \frac{AB}{AC} = \frac{BD}{DC} \]

### Step 3: Calculate the Ratios
Now, we will calculate the ratios using the given lengths:
- \( AB = 6 \) cm
- \( AC = 8 \) cm
- \( BD = 3 \) cm
- \( DC = 4 \) cm

Calculating the left side of the equation:

\[ \frac{AB}{AC} = \frac{6}{8} = \frac{3}{4} \]

Now, calculating the right side of the equation:

\[ \frac{BD}{DC} = \frac{3}{4} \]

### Step 4: Compare the Ratios
Since both ratios are equal:

\[ \frac{AB}{AC} = \frac{BD}{DC} \implies \frac{3}{4} = \frac{3}{4} \]

This confirms that the assertion is correct.

### Step 5: Analyze the Reason
The reason states that the angle bisector AD divides the base BC in the ratio of AB to AC. Since we have already established that the angle bisector divides the opposite side in the ratio of the adjacent sides, the reason correctly explains the assertion.

### Conclusion
Both the assertion and the reason are true, and the reason is indeed the correct explanation for the assertion.

### Final Answer
- **Assertion (A)**: True
- **Reason (R)**: True
- **Explanation**: The angle bisector divides the opposite side in the ratio of the adjacent sides.

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