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October 8th, 2019, 11:30 AM   #1
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find the coordinates of the 2 sides of a chord

If I have the circle's equation including r and circle's origin coordinates. And I have the coordinates of a point M in the middle of a chord which splits the chord into 2 equal parts. Then how do I find the coordinates of each point in each side of the chord?
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October 8th, 2019, 12:18 PM   #2
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Let $c$ be the half-chord (distance from M to either end of the chord). Let $l$ be the distance between M and the origin O of the circle. Then, by the Pythagorean theorem,
$l^2 + c^2 = r^2$.
You know $l$ and $r$, so you can find $c$. Now just go $c$ units from M in either direction 90° from segment OM.

Hopefully this helped. If not, draw a diagram, and we can walk you through it further.
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October 9th, 2019, 02:03 AM   #3
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Let the two points be A and B. Try to apply: $\displaystyle d(A,B)=2\cdot d(A,M)=2\cdot d(B,M)$.
$\displaystyle d(A,B)$ is the distance between A and B.
$\displaystyle d(A,M) \cdot d(M,B) = x(r-x)\:$ , where x is the minimal distance between M and the circle .
Now take examples and verify the result you got with a graphing calculator .

Last edited by idontknow; October 9th, 2019 at 02:27 AM.
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