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The line AB is 6 m, in length and is tangent to the inner of the two concentric circles at point C. It is known that the radii of the two circles are integers. The radiusof the outer circle is-------, where A and B are points on the outer circle.
- a)5 m
- b)4 m
- c)6 m
- d)3 m
Correct answer is option 'A'. Can you explain this answer?
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The line AB is 6 m, in length and is tangent to the inner of the two c...
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Given Information
- The line AB is 6 m in length and is tangent to the inner of the two concentric circles at point C.
- The radii of the two circles are integers.
To Find
- The radius of the outer circle.
Solution
Let's consider the diagram below:
We can see that the radius of the inner circle is equal to the length of the perpendicular from the center of the circles to the line AB.
Let O be the center of the circles, and let OC be x. Then, by the Pythagorean theorem, we have:
OA^2 = OC^2 + AC^2
OB^2 = OC^2 + BC^2
Since OA = OB (both A and B are on the outer circle), we can subtract the two equations to get:
AC^2 - BC^2 = 0
(AC + BC)(AC - BC) = 0
Since AC and BC are both positive, we have AC = BC.
Therefore, the line AB is equidistant from A and B, which means it passes through the center O of the circles.
Let r be the radius of the outer circle. Then, we have:
OC = r - x (since x is the radius of the inner circle)
By the Pythagorean theorem, we also have:
AC^2 = r^2 - OC^2 = r^2 - (r - x)^2 = 2rx - x^2
Since AC = BC, we have:
2rx - x^2 = (6/2)^2 = 9
Simplifying, we get:
x(2r - x) = 9
Since x is an integer, and 2r - x is also an integer, we can see that x must be a factor of 9. The possible values of x are:
x = 1, 3, or 9
If x = 1, then 2r - x = 2r - 1 is odd, which means r is not an integer.
If x = 3, then 2r - x = 2r - 3 is odd, which means r is not an integer.
Therefore, we must have x = 9, which gives:
2r - x = 2r - 9
x(2r - x) = 9
Substituting x = 9, we get:
r = (x^2 + 9)/2x = (81 + 9)/18 = 5
Therefore, the radius of the outer circle is 5 m.
Answer: (a) 5 m
- The line AB is 6 m in length and is tangent to the inner of the two concentric circles at point C.
- The radii of the two circles are integers.
To Find
- The radius of the outer circle.
Solution
Let's consider the diagram below:
We can see that the radius of the inner circle is equal to the length of the perpendicular from the center of the circles to the line AB.
Let O be the center of the circles, and let OC be x. Then, by the Pythagorean theorem, we have:
OA^2 = OC^2 + AC^2
OB^2 = OC^2 + BC^2
Since OA = OB (both A and B are on the outer circle), we can subtract the two equations to get:
AC^2 - BC^2 = 0
(AC + BC)(AC - BC) = 0
Since AC and BC are both positive, we have AC = BC.
Therefore, the line AB is equidistant from A and B, which means it passes through the center O of the circles.
Let r be the radius of the outer circle. Then, we have:
OC = r - x (since x is the radius of the inner circle)
By the Pythagorean theorem, we also have:
AC^2 = r^2 - OC^2 = r^2 - (r - x)^2 = 2rx - x^2
Since AC = BC, we have:
2rx - x^2 = (6/2)^2 = 9
Simplifying, we get:
x(2r - x) = 9
Since x is an integer, and 2r - x is also an integer, we can see that x must be a factor of 9. The possible values of x are:
x = 1, 3, or 9
If x = 1, then 2r - x = 2r - 1 is odd, which means r is not an integer.
If x = 3, then 2r - x = 2r - 3 is odd, which means r is not an integer.
Therefore, we must have x = 9, which gives:
2r - x = 2r - 9
x(2r - x) = 9
Substituting x = 9, we get:
r = (x^2 + 9)/2x = (81 + 9)/18 = 5
Therefore, the radius of the outer circle is 5 m.
Answer: (a) 5 m
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