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The angle subtended by the chord of length 10 cm is 120° at the centre. Calculate the distance of the chord (in cm) from the centre.
- a)5/√3
- b)6/√3
- c)4/√3
- d)5/(2√3)
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
The angle subtended by the chord of length 10 cm is 120° at the ce...
Given, ∠AOB = 120° & AB = 10 cm
Draw OD ⊥ AB
So, ∠DOB = 1/2 × ∠AOB = 1/2 × 120 = 60°
⇒ DB = 1/2 × 10 = 5 cm
Now, In ΔODB
⇒ cot 60° = OD/DB
⇒ 1/(√3) = OD/5
⇒ OD = 5/(√3)
Therefore, distance of the chord from the centre is 5/(√3)
Most Upvoted Answer
The angle subtended by the chord of length 10 cm is 120° at the ce...
Given Data
- Chord length (c) = 10 cm
- Angle subtended at the center (θ) = 120°
Understanding the Geometry
To find the distance of the chord from the center of the circle, we can use the relationship between the chord length, the radius, and the angle subtended at the center.
Step 1: Calculate the radius (R)
Using the formula for the length of a chord:
\[ c = 2R \sin\left(\frac{\theta}{2}\right) \]
Substituting the known values:
\[ 10 = 2R \sin\left(\frac{120°}{2}\right) \]
This simplifies to:
\[ 10 = 2R \sin(60°) \]
Since \( \sin(60°) = \frac{\sqrt{3}}{2} \):
\[ 10 = 2R \cdot \frac{\sqrt{3}}{2} \]
Thus:
\[ 10 = R\sqrt{3} \]
So,
\[ R = \frac{10}{\sqrt{3}} \]
Step 2: Calculate the distance (d) from the center to the chord
Using the relationship:
\[ d = R \cos\left(\frac{\theta}{2}\right) \]
Substituting the values:
\[ d = \frac{10}{\sqrt{3}} \cos(60°) \]
Since \( \cos(60°) = \frac{1}{2} \):
\[ d = \frac{10}{\sqrt{3}} \cdot \frac{1}{2} \]
Thus:
\[ d = \frac{5}{\sqrt{3}} \]
Conclusion
The distance of the chord from the center is:
Answer: \(\frac{5}{\sqrt{3}} \, \text{cm}\)
This matches with option 'A'.
- Chord length (c) = 10 cm
- Angle subtended at the center (θ) = 120°
Understanding the Geometry
To find the distance of the chord from the center of the circle, we can use the relationship between the chord length, the radius, and the angle subtended at the center.
Step 1: Calculate the radius (R)
Using the formula for the length of a chord:
\[ c = 2R \sin\left(\frac{\theta}{2}\right) \]
Substituting the known values:
\[ 10 = 2R \sin\left(\frac{120°}{2}\right) \]
This simplifies to:
\[ 10 = 2R \sin(60°) \]
Since \( \sin(60°) = \frac{\sqrt{3}}{2} \):
\[ 10 = 2R \cdot \frac{\sqrt{3}}{2} \]
Thus:
\[ 10 = R\sqrt{3} \]
So,
\[ R = \frac{10}{\sqrt{3}} \]
Step 2: Calculate the distance (d) from the center to the chord
Using the relationship:
\[ d = R \cos\left(\frac{\theta}{2}\right) \]
Substituting the values:
\[ d = \frac{10}{\sqrt{3}} \cos(60°) \]
Since \( \cos(60°) = \frac{1}{2} \):
\[ d = \frac{10}{\sqrt{3}} \cdot \frac{1}{2} \]
Thus:
\[ d = \frac{5}{\sqrt{3}} \]
Conclusion
The distance of the chord from the center is:
Answer: \(\frac{5}{\sqrt{3}} \, \text{cm}\)
This matches with option 'A'.
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The angle subtended by the chord of length 10 cm is 120° at the centre. Calculate the distance of the chord (in cm) from the centre.a)5/√3b)6/√3c)4/√3d)5/(2√3)Correct answer is option 'A'. Can you explain this answer?
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