Intersecting Circles

cmdrpaddy

I found a question in my maths book that i can't figure out at all. it goes like this: If two circles of radii a and b intersect at an acute angle theta show that the lenght of their common chord is :

(2*a*b*Sin(theta))/(sqrt(a^2+b^2+2*a*b*Cos(theta)))

any help would be appreciated... even a clue...

silverside

ok maybe this is spelling it out too much but:

start out by drawing a triangle between the intersection of the circles and their two centres.

The angle the circles (ie their tangents) make is the same as the angle their radii make at the point of contact as the tangent is perpendicular to the radius.

So know you have a triangle, two sides a and b, and the angle at the vertex theta.
You want to get the height times 2.

You can then either work from first principles using coordinate geometry and pythagoras theorem, or if you know the formulae you can use them, to get the area and 3rd side of a triangle given 2 sides and an angle between them. Once you get the length of the 3rd side remember the area is also 1/2 base by height so height is 2*area/base. Remember sin^2+cos^2=1 as this will simlify the calculations.

After a page or so of manipulations you should get the answer given.

chakotha

I'm not sure what you mean by intersecting at angle theta.

Is it the angle at an intersection between the radii?

ecksor

The angle between the tangents to that point.

chakotha

It's like a proof of the Sine rule!

cmdrpaddy

thanks for the help, my brain's been like mush lately can't seem to think straight. thasnks for the help:)