AIME 2004, #10: prob. that circle does not meet diagonal

[malick]

#10 (AIME 04) states:

A circle radius 1 is randomly placed so that it lies entirely inside a 15 x 36 rectangle ABCD. Find the probability that it does not meet the diagonal AC.

This is the answer, can anyone explain it to me?

If the circle lies entirely inside the rectangle, then its center must lie inside a 13×34 rectangle, area 442. If it does not intersect the diagonal, then its center must lie inside one of the two triangles shown. Note that a 5-12-13 triangle has inradius 2 (because if the inradius is r, then calculating the area two ways we get 5·12/2 = r(5+12+13)/2). So the triangles are 12½, 30, 32½. Thus their area is 30·12½ = 375. So the probability is 375/442.

 

[stapel]

It might help if you said where you're stuck.

For instance, you know how the find the area of a rectangle, so that part is okay, right? But are you not familiar with the formulae and rules regarding inradii and such?

When you reply, please describe the picture that you're looking at. Thank you.

Eliz.

 

[malick]

thanks, I finally get it. I just didn't know how to calculate the inradius.

 

[malick]

correction, there is still one thing that I do not get, why then must its center lie inside a 13×34 rectangle.

Thanks

 

[pka]

correction, there is still one thing that I do not get, why then must its center lie inside a 13×34 rectangle.

Otherwise the circle would not be in the interior of the original rectangle.