In a circle with centre P,a chord AB is parallel to a tangent & inersects the radius drawn from a point of contact at the midpoint of the radius .If AB=12,FIND THE RADIUS OF THE CIRCLE.
let the radius be 2x
then I see a right-angled triangle with sides x , 6 and hypotenuse 2x
(2x)^2 = x^2 + 6^2
3x^2 = 36
x^2 = 12
x = √12 = 2√3
so the radius is 4√3
To find the radius of the circle, we can use the properties of a circle.
Given:
- Chord AB is parallel to a tangent.
- The chord intersects the radius drawn from a point of contact at the midpoint of the radius.
- Length of chord AB is 12 units.
Step 1: Draw a diagram
- Draw a circle with center P.
- Draw a radius line from P to any point of contact on the circle.
- Draw tangent line from point of contact to a point where chord AB intersects the radius.
- Label chord AB with length 12.
Step 2: Use the properties of a tangent and a chord
- In a circle, a tangent line drawn to a point on the circle is perpendicular to the radius at that point.
- Since AB is parallel to the tangent, it means AB is also perpendicular to the radius.
- Connecting the center, P, to the midpoint of the radius, we have a right-angled triangle with one side being the radius (RP) and the other side being half the length of AB (6 units).
Step 3: Apply Pythagoras' Theorem
- In a right-angled triangle, the square of the hypotenuse (RP) is equal to the sum of the squares of the other two sides.
- Applying Pythagoras' Theorem, we have: RP^2 = (6 units)^2 + r^2,
where r is the radius of the circle.
Step 4: Substitute the known values and solve for r
- We know that AB = 12 units, and the radius is r.
- Substituting the values, we have: r^2 = (6 units)^2 + (r units)^2
r^2 = 36 + r^2
0 = 36
- We reach a contradiction since the equation is impossible.
- Hence, there is no valid solution for the radius of the circle that satisfies the given conditions.
Conclusion:
- Based on the given conditions, there is no valid value for the radius of the circle that satisfies the properties mentioned.
To find the radius of the circle, we can use the properties of a tangent and a chord.
First, let's draw a diagram of the situation described in the question.
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B
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________|________ <- Circle with center P
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P
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A
```
From the information given, we know that AB is parallel to a tangent, and the chord AB intersects the radius drawn from a point of contact at the midpoint of the radius.
Let the point of contact be M, and the midpoint of the radius be O.
```
B
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________|________ <- Circle with center P
|M |
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P--------O
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A
```
Since AB is parallel to the tangent, we can use the properties of parallel lines to find the relationship between the angles.
The angle between the chord AB and the tangent at point M is equal to the angle formed by the radius OP and the chord AB.
Let's call the angle between AB and the tangent at point M as theta.
Now, we have a right triangle OMP, where OP is the radius of the circle.
Since M is the midpoint of the radius and AB is the chord passing through M, we can consider triangle AMB as an isosceles triangle. Therefore, AM = BM = AB/2 = 12/2 = 6 units.
We need to find the radius OP.
Here's how we can find it:
1. Find the length of OA.
Since AMB is an isosceles triangle, OM is perpendicular to AB and bisects it. So, in triangle OMA, OM is the height and AM is the base of the triangle.
Using the Pythagorean theorem, we have:
OA^2 = OM^2 + AM^2
OA^2 = (OP - PM)^2 + AM^2
OA^2 = (OP - OP/2)^2 + 6^2
OA^2 = (OP/2)^2 + 36
OA^2 = OP^2/4 + 36
2. Find the length of OA.
Since AB is parallel to the tangent, the angle between AB and the tangent is theta.
Using the property of circles, we know that the angle between the chord AB and the tangent at any point of contact is equal to the angle formed between the radius drawn from the center to the point of contact and the chord.
Therefore, we have:
tan(theta) = OA/AM
tan(theta) = OA/6
OA = 6 * tan(theta)
3. Equate the two expressions for OA found in step 1 and step 2 and solve for OP.
OP^2/4 + 36 = (6 * tan(theta))^2
OP^2/4 + 36 = 36 * tan^2(theta)
OP^2/4 = 36 * tan^2(theta) - 36
OP^2/4 = 36(tan^2(theta) - 1)
OP^2 = 4 * 36(tan^2(theta) - 1)
OP^2 = 144(tan^2(theta) - 1)
OP^2 = 144 * tan^2(theta) - 144
OP^2 = 144 * tan^2(theta) - 144
OP^2 = 144 * tan^2(theta) - 144
OP^2 = 144 * tan^2(theta) - 144
OP^2 = 144 * tan^2(theta) - 144
OP = sqrt(144 * tan^2(theta) - 144)
Finally, you can substitute the value of theta into the equation to find the radius OP.