In triangle ABC, AB = 12 cm, `angle B = 60^@`, the perpendicular from A to BC meets it at D. The bisector of `angle ABC` meets AD at E. Then E divides AD in the ratio

Options

A. `3 :1`
B. `6 :1`
C. `1 :1`
D. `2 :1`

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the Triangle
We have triangle ABC with:
- AB = 12 cm
- Angle B = 60 degrees

### Step 2: Draw the Perpendicular
A perpendicular line is drawn from point A to line BC, meeting at point D. This means that angle ADB is a right angle (90 degrees).

### Step 3: Use Cosine to Find AD
In triangle ABD, we can use the cosine of angle B to find the length of AD.
- Cosine of angle B (60 degrees) is given by:
\[ \cos(60^\circ) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{AD}{AB} \]
- Substituting the known values:
\[ \cos(60^\circ) = \frac{AD}{12} \]
- Since \(\cos(60^\circ) = \frac{1}{2}\), we have:
\[ \frac{1}{2} = \frac{AD}{12} \]
- Solving for AD:
\[ AD = 12 \times \frac{1}{2} = 6 \text{ cm} \]

### Step 4: Apply the Angle Bisector Theorem
According to the angle bisector theorem, the ratio in which point E divides AD is given by:
\[ \frac{AE}{ED} = \frac{AB}{BD} \]
- Here, AB = 12 cm and we need to find BD.

### Step 5: Find BD
In triangle ABD, since AD is the height and we already found AD = 6 cm, we can find BD using the sine of angle B:
- We know that:
\[ \sin(60^\circ) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{BD}{AB} \]
- Substituting the known values:
\[ \sin(60^\circ) = \frac{BD}{12} \]
- Since \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\), we have:
\[ \frac{\sqrt{3}}{2} = \frac{BD}{12} \]
- Solving for BD:
\[ BD = 12 \times \frac{\sqrt{3}}{2} = 6\sqrt{3} \text{ cm} \]

### Step 6: Calculate the Ratio AE/ED
Now we can substitute the values into the angle bisector theorem:
\[ \frac{AE}{ED} = \frac{AB}{BD} = \frac{12}{6\sqrt{3}} = \frac{2}{\sqrt{3}} \]
To express this ratio in a simpler form, we can rationalize the denominator:
\[ \frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3} \]

### Step 7: Final Ratio
Thus, the ratio AE:ED can be expressed as:
\[ AE:ED = 2:3 \]

### Conclusion
The final answer is that E divides AD in the ratio of 2:1.