Determine angle of intersecting lines inside a circle

2milehi

So I ran across this problem on the 'net and I can't determine "x". The arc length of the circle is 360.



I added some other variable and took what I know about a circle and intersecting lines. I wound up with four variables and four equations.

x = 1/2 (y + 67)
w = 1/2 (z + 147)
y + z + 67 + 147 = 360
2w + 2x = 360

and into matrix form

1w + 1x + 0y + 0z = 180
0w + 1x - 1/2y + 0z = 67/2
1w + 0x - 0y - 1/2z = 147/2
0w + 0x + 1y + 1z = 146

But that comes up with an indeterminate.

Taking a closer look before I post, I see that three of the equations relate to length and one relates to degrees. But with s = r · theta, r is such that s = theta in degrees.

I am stuck now

HallsofIvy

The reason you get "indeterminate" is that those four equations are not independent. And the problem itself does not have enough information. You could move that pretty much any where around the circle changing y and z but not x and w.

(Since you say "the arclength of the circle is 360" I suspect that y and z are in "degrees of arc", not length.)

2milehi

There is an answer given for it and it does work out for all angles and degree of arc. So there should be a way to figure it out, hence why there is a measure of 147.

2milehi

Can anyone else figure it out?

2milehi

It took a bit to sink in, but now I understand. There are an infinite number of solutions because of the lack of information.