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The radii of two concentric circles are 13 cm and 8 cm. AB is a diameter of the bigger circle and BD is tangent at the smaller circle touching it at D and the bigger circle at E . point A is joined to D . the length of AD is?
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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.
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.
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The radii of two concentric circles are 13 cm and 8 cm. AB is a diameter of the bigger circle and BD is tangent at the smaller circle touching it at D and the bigger circle at E . point A is joined to D . the length of AD is?
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The radii of two concentric circles are 13 cm and 8 cm. AB is a diameter of the bigger circle and BD is tangent at the smaller circle touching it at D and the bigger circle at E . point A is joined to D . the length of AD is? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The radii of two concentric circles are 13 cm and 8 cm. AB is a diameter of the bigger circle and BD is tangent at the smaller circle touching it at D and the bigger circle at E . point A is joined to D . the length of AD is? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The radii of two concentric circles are 13 cm and 8 cm. AB is a diameter of the bigger circle and BD is tangent at the smaller circle touching it at D and the bigger circle at E . point A is joined to D . the length of AD is?.
The radii of two concentric circles are 13 cm and 8 cm. AB is a diameter of the bigger circle and BD is tangent at the smaller circle touching it at D and the bigger circle at E . point A is joined to D . the length of AD is? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The radii of two concentric circles are 13 cm and 8 cm. AB is a diameter of the bigger circle and BD is tangent at the smaller circle touching it at D and the bigger circle at E . point A is joined to D . the length of AD is? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The radii of two concentric circles are 13 cm and 8 cm. AB is a diameter of the bigger circle and BD is tangent at the smaller circle touching it at D and the bigger circle at E . point A is joined to D . the length of AD is?.
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