The radii of two concentric circles are 13 cm and 8 cm. AB is a diamet...
Understanding the Geometry
The problem involves two concentric circles with radii 13 cm (larger circle) and 8 cm (smaller circle). We will determine the length of segment AD where AB is the diameter of the larger circle, and BD is a tangent to the smaller circle.
Key Points
- Radius of Larger Circle (R): 13 cm
- Radius of Smaller Circle (r): 8 cm
- Diameter AB: Since AB is a diameter of the larger circle, its length is 2R = 26 cm.
Positioning Points
- Let the center of both circles be O.
- Point A lies on the circumference of the larger circle, while point D is where the tangent BD touches the smaller circle.
- Since BD is tangent to the smaller circle, the radius OD is perpendicular to BD.
Applying the Pythagorean Theorem
In triangle OBD:
- OB = R = 13 cm (radius of larger circle)
- OD = r = 8 cm (radius of smaller circle)
- BD is tangent, so we apply the Pythagorean theorem:
Calculating Length BD
Using the theorem:
\[
OB^2 = OD^2 + BD^2
\]
Substituting the known values:
\[
13^2 = 8^2 + BD^2
\]
\[
169 = 64 + BD^2
\]
\[
BD^2 = 169 - 64 = 105
\]
Thus,
\[
BD = \sqrt{105} \approx 10.25 \, \text{cm}
\]
Finding AD
In triangle AOD:
- AD = AO + OD
- AO (the radius of the larger circle) = 13 cm
- OD (the radius of the smaller circle) = 8 cm
Hence,
\[
AD = AO + OD = 13 + 8 = 21 \, \text{cm}
\]
Final Answer
The length of AD is 21 cm.