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Chord Length of Intersecting Circles

chdlen.html

Exploration

"chdlen_1.gif"

Graphics saved as "cir19.eps".

Show that the distance, d, between the intersection points of two circles is given by

"chdlen_2.gif"

where D is the distance between the centers of the circles, and "chdlen_3.gif" and "chdlen_4.gif" are the radii of the two circles.

Approach

Assume the radii of the two circles centered at "chdlen_5.gif" and "chdlen_6.gif" are "chdlen_7.gif" and "chdlen_8.gif", respectively, "chdlen_9.gif" is one of the intersection points, and the distance between the centers is D. The length of the common chord, d, can be found by equating the area (squared) of "chdlen_10.gif" using Heron's formula and the standard area formula A=b h/2.

Initialize

To initialize Descarta2D, select the input cell bracket and press SHIFT-Enter.

This initialization assumes that the Descarta2D software has been copied into one of the standard directories for AddOns which are on the Mathematica search path, $Path.

<<Descarta2D`

Solution

"chdlen_11.gif" is the area (squared) by Heron's formula.

Clear[r1,r2,D1];
s=(r1+r2+D1)/2;
A1=s(s-r1)(s-r2)(s-D1) //Simplify

"chdlen_12.gif"

"chdlen_13.gif" is the area (squared) by the standard area formula A=b h/2 (d is the distance between the intersection points, i.e. the length of the chord).

Clear[d];
A2=(D1*(d/2)/2)^2 //Simplify

"chdlen_14.gif"

Set the areas equal and solve for d. Take the positive value. Simplify the result involving "chdlen_15.gif" with an algebraically equivalent expression.

ans=Solve[A1==A2,d]

"chdlen_16.gif"

"chdlen_17.gif"

"chdlen_18.gif"

Discussion

If the radii are equal the result can be significantly simplified.

Clear[r];
ans2=ans1 //. {r1->r, r2->r}

"chdlen_19.gif"


Copyright © 1999-2007 Donald L. Vossler, Descarta2D Publishing
www.Descarta2D.com