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The distance between the centers of the two circles is 61cm and their radius is 35cm and 24cm. What is the length(in cm) of the direct common tangent to the circles?
- a)60
- b)54
- c)48
- d)72
Correct answer is option 'A'. Can you explain this answer?
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The distance between the centers of the two circles is 61cm and their...
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Understanding the Problem
To find the length of the direct common tangent between two circles, we can use the formula:
Length of Direct Common Tangent
\[ L = \sqrt{d^2 - (r_1 - r_2)^2} \]
Where:
- \( L \) is the length of the tangent
- \( d \) is the distance between the centers of the circles
- \( r_1 \) and \( r_2 \) are the radii of the circles
Given in the problem:
- Distance between centers (\( d \)) = 61 cm
- Radius of the first circle (\( r_1 \)) = 35 cm
- Radius of the second circle (\( r_2 \)) = 24 cm
Calculating the Difference of Radii
First, we calculate the difference of the radii:
- \( r_1 - r_2 = 35 - 24 = 11 \) cm
Applying the Formula
Now, substitute the values into the formula:
\[ L = \sqrt{61^2 - 11^2} \]
Calculating the squares:
- \( 61^2 = 3721 \)
- \( 11^2 = 121 \)
Now, substitute these values:
\[ L = \sqrt{3721 - 121} = \sqrt{3600} \]
Finding the Final Length
Calculating the square root:
- \( L = 60 \) cm
Thus, the length of the direct common tangent to the circles is **60 cm**.
Conclusion
The correct answer is option **A**: **60 cm**.
To find the length of the direct common tangent between two circles, we can use the formula:
Length of Direct Common Tangent
\[ L = \sqrt{d^2 - (r_1 - r_2)^2} \]
Where:
- \( L \) is the length of the tangent
- \( d \) is the distance between the centers of the circles
- \( r_1 \) and \( r_2 \) are the radii of the circles
Given in the problem:
- Distance between centers (\( d \)) = 61 cm
- Radius of the first circle (\( r_1 \)) = 35 cm
- Radius of the second circle (\( r_2 \)) = 24 cm
Calculating the Difference of Radii
First, we calculate the difference of the radii:
- \( r_1 - r_2 = 35 - 24 = 11 \) cm
Applying the Formula
Now, substitute the values into the formula:
\[ L = \sqrt{61^2 - 11^2} \]
Calculating the squares:
- \( 61^2 = 3721 \)
- \( 11^2 = 121 \)
Now, substitute these values:
\[ L = \sqrt{3721 - 121} = \sqrt{3600} \]
Finding the Final Length
Calculating the square root:
- \( L = 60 \) cm
Thus, the length of the direct common tangent to the circles is **60 cm**.
Conclusion
The correct answer is option **A**: **60 cm**.
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The distance between the centers of the two circles is 61cm and their radius is 35cm and 24cm. What is the length(in cm) of the direct common tangent to the circles?a)60b)54c)48d)72Correct answer is option 'A'. Can you explain this answer?
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