# A chord AB of length 2 cm is drawn in a circle of radius 3 cm. A diameter BC is constructed, and the tangent from C is drawn. The chord AB is extended to meet the tangent at D. Find the length of AD.

## To find the length of AD, we can use the properties of tangent and chords in a circle.

Step 1: Draw the figure as described in the problem.

```
A
-
/ \
/ \
/ \
B C
- |
/ |
/ D
```

Step 2: Identify relevant geometric relationships.

- In a circle, a tangent is always perpendicular to the radius drawn to the point of tangency. Therefore, angle BCD is 90 degrees.
- Chord AB is extended to meet the tangent at point D. Angle ABC and angle BCD are vertical angles and thus congruent.
- We know the lengths of chord AB (2 cm) and radius BC (3 cm).

Step 3: Apply the properties to solve the problem.

Since angle ABC is congruent to angle BCD, we conclude that angle ABC is also 90 degrees. Therefore, AB is a diameter of the circle.

The length of a diameter is twice the length of the radius. Hence, we can determine that AB = 6 cm.

Now, we have a right triangle ABD with sides AD, AB, and BD. We can use the Pythagorean theorem to find the length of AD.

Pythagorean theorem: In a right triangle, the square of the hypotenuse (AB) is equal to the sum of the squares of the other two sides (AD and BD).

Substituting the known values: AB = 6 cm and BD = 3 cm, we get:

Simplifying:

Subtracting 9 from both sides: