Calculate Circle Diameter Given Two Chords: AX=BX=6 cm, CX=4 cm, DX=9 cm

It seems like the setup described is exactly what the problem is asking for. In summary, the conversation is about calculating the radius of a circle given two intersecting chords and their respective lengths. One person suggests constructing a secant and tangent to find the solution, while another person disagrees and believes there is not enough information provided. They discuss the possibility of finding the smallest circle and its corresponding radius, but ultimately agree that the given information is insufficient for a definitive answer.
  • #1
John O' Meara
330
0
Two chords of a circle, AB and CD, intersect at X, such that AX=BX= 6 cm. CX=4 cm and DX=9 cm. Calculate the radius of the circle, given that it has an integer value. I constructed a secant EG of a diameter FG of the circle and drew a tangent from C to it at E, then we have from the intersecting secant theroem: EFxEG=EC^2.Many thanks guys.
 
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  • #2
I don't understand your work.
 
  • #3
All I did was to draw a tangent at point C, to the circle, then drew a diameter FG of the circle such that, when you produce the diameter GF it meets the tangent at E.Sorry for the long delay in reply. Many thanks.
 
  • #4
Is FG just any diamater that isn't parallel to the tangent at C? And I don't see why you did what you did.
 
  • #5
The reason why I have done what I did, was that this question is posed in the section of a book called "intersecting chord and secant theorems".Even though I would need to know the length EF and the tangent |EC|. Since I don't the solution to the problem eludes me.Maybe someone could start me off!
 
  • #6
The radius is quite guessable from observing that the lengths of the 4 segments satisfy Archimedis' lemma for perpendicular chords.
 
  • #7
The chords aren't necessarily perpendicular (are they?).
 
  • #8
No, the chords aren't necessarily perpendicular. The answer is 10cm. The question is how did they get it!
 
  • #9
I don't think there's enough information here. Take any circle of sufficiently large size (this will be clear in a minute). Draw a chord of length 12, and call the endpoints A and B. Then draw a line from the center of this chord X, to a point C on on the arc between A and B, such that the length of CX is 4. Note this is possible since the maximum length such a segment could have is clearly 6, and the minimum can be made arbitrarily small by increasing the radius of the circle. Now extend this line to intersect the circle again at a point D, and since AX*BX=CX*DX, we must have DX=9. Am I misssing something, or is this exactly the setup you've described? Maybe you're asked to find the smallest such circle?
 
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  • #10
StatusX said:
I don't think there's enough information here. Take any circle of sufficiently large size (this will be clear in a minute). Draw a chord of length 12, and call the endpoints A and B. Then draw a line from the center of this chord X, to a point C on on the arc between A and B, such that the length of CX is 4. Note this is possible since the maximum length such a segment could have is clearly 6, and the minimum can be made arbitrarily small by increasing the radius of the circle. Now extend this line to intersect the circle again at a point D, and since AX*BX=CX*DX, we must have DX=9. Am I misssing something, or is this exactly the setup you've described? Maybe you're asked to find the smallest such circle?

But then the smallest such circle would be one in which the longest chord is diameter, andthe radius would be equal to 6.5.

I agree, there is not enough information.
 

What is the formula for calculating the diameter of a circle given two chords?

The formula for calculating the diameter of a circle given two chords is:
D = √[(4 * a * b * c)/(a + b + c)], where a, b, and c are the lengths of the chords.

Can this formula be used for any two chords in a circle?

Yes, this formula can be used for any two chords in a circle as long as the lengths of the chords are known.

How do I know which chord is chord "a", "b", or "c" in the formula?

In this formula, "a" and "b" represent the lengths of the two equal chords (AX=BX=6 cm), while "c" represents the length of the third chord (CX=4 cm or DX=9 cm).

What if I am given three equal chords instead of two?

If you are given three equal chords, you can use the formula:
D = √[(2 * a * b * c)/(a + b + c)], where a, b, and c are the lengths of the chords.

Can I use this formula to find the diameter of a circle given only one chord and the radius?

Yes, you can use this formula by first finding the length of the other chord using the Pythagorean theorem. Then, you can use the formula:
D = √[(4 * a * b * c)/(a + b + c)], where a is the length of the known chord, b is the length of the other chord, and c is the radius of the circle.

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