In `DeltaABC /_A = /_B = 60^@, AC= sqrt(13) cm`. The lines `AD and BD `intersect at D with `/_D = 90^@`. If `DB = 2 cm`, then the length of AD is

Options

A. 3 cm
B. 3.5 cm
C. 4 cm
D. 4.7 cm

Answer

To solve the problem step by step, we will analyze the given information and apply the relevant geometric principles.

### Step-by-Step Solution:

1. **Understanding the Triangle**: We have triangle \( ABC \) with angles \( \angle A = \angle B = 60^\circ \). This means \( \triangle ABC \) is an isosceles triangle, and since the sum of angles in a triangle is \( 180^\circ \), it follows that \( \angle C = 60^\circ \) as well. Therefore, triangle \( ABC \) is equilateral.

2. **Identifying the Sides**: The side \( AC \) is given as \( \sqrt{13} \) cm. Since triangle \( ABC \) is equilateral, all sides are equal. Thus, \( AB = AC = BC = \sqrt{13} \) cm.

3. **Drawing the Perpendiculars**: We draw lines \( AD \) and \( BD \) such that they intersect at point \( D \) with \( \angle D = 90^\circ \). We are given that \( DB = 2 \) cm.

4. **Using Properties of the Triangle**: In an equilateral triangle, if a perpendicular is drawn from a vertex to the opposite side, it bisects that side. Thus, since \( D \) is the foot of the perpendicular from \( A \) to \( BC \), we can conclude that \( DC = DB = 2 \) cm.

5. **Calculating \( BC \)**: The total length of side \( BC \) can be calculated as:
\[ BC = DB + DC = 2 \text{ cm} + 2 \text{ cm} = 4 \text{ cm} \]

6. **Applying the Pythagorean Theorem**: Now, we will focus on triangle \( ADC \), which is a right triangle. We can apply the Pythagorean theorem:
\[ AC^2 = AD^2 + DC^2 \]
Substituting the known values:
\[ (\sqrt{13})^2 = AD^2 + (2)^2 \]
This simplifies to:
\[ 13 = AD^2 + 4 \]

7. **Solving for \( AD \)**: Rearranging the equation gives:
\[ AD^2 = 13 - 4 = 9 \]
Taking the square root of both sides:
\[ AD = \sqrt{9} = 3 \text{ cm} \]

### Final Answer:
The length of \( AD \) is \( 3 \) cm.

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