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.
by "easy enough" ole means that you have similar right-angled triangles, so use ratios.
Can someone show the work? I'm also stuck on this question.
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).
AB^2 = AD^2 + BD^2
Substituting the known values: AB = 6 cm and BD = 3 cm, we get:
6^2 = AD^2 + 3^2
Simplifying:
36 = AD^2 + 9
Subtracting 9 from both sides:
27 = AD^2
Taking the square root of both sides:
√27 = √(AD^2)
√27 = AD
Therefore, the length of AD is √27 cm, which simplifies to 3√3 cm or approximately 5.196 cm.