SOLUTION: In rectangle ABCD, AB = 8 and BC = 20. Let P be a point on AD such that 6*angle BPC = 90◦. If r1, r2, r3 are the radii of the incircles of triangles APB, BPC and CPD, what i

Algebra ->  Circles -> SOLUTION: In rectangle ABCD, AB = 8 and BC = 20. Let P be a point on AD such that 6*angle BPC = 90◦. If r1, r2, r3 are the radii of the incircles of triangles APB, BPC and CPD, what i      Log On


   

Question 1090909: In rectangle ABCD, AB = 8 and BC = 20. Let P be a point on AD such that 6*angle BPC = 90◦.
If r1, r2, r3 are the radii of the incircles of triangles APB, BPC and CPD, what is the value
of r1 + r2 + r3?
Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!

Your statement of the problem appears to say that angle BPC has to be 15 degrees. (The way I see the problem, you say 6 times the angle measure is 90, making the angle 15.)

With the given dimensions of the rectangle, there is no point P on AD that will make angle BPC 15 degrees.

But the problem is a very cool problem if the statement of the problem is supposed to be that the measure of angle BPC is 90 degrees. So I will assume that was the intention, and I will show a solution for that problem.


If angle BPC is indeed a right angle, then triangles PAB, BPC, and CDP are all similar. We can use those similarities to find the length of AP. If we call the length of AP x, then, comparing corresponding sides of triangles PAB and CDP, we have

8%2Fx+=+%2820-x%29%2F8
64+=+20x-x%5E2
x%5E2-20x%2B64+=+0
%28x-4%29%28x-16%29+=+0

So AP is 4, and PD is 16.

Now we can find the inradius of triangle PAB using the formula
A+=+%281%2F2%29pr
where A is the area of the triangle, p is the perimeter of the triangle, and r is the inradius.

%281%2F2%29%284%29%288%29+=+%281%2F2%29%2812%2B4%2Asqrt%285%29%29r
r+=+16%2F%2812%2B4%2Asqrt%285%29%29
r+=+4%2F%283%2Bsqrt%285%29%29
r+=+%284%283-sqrt%285%29%29%29%2F%28%283%2Bsqrt%285%29%29%2A%283-sqrt%285%29%29%29
r+=+4%283-sqrt%285%29%29%2F4
r+=+3-sqrt%285%29

We could do similar calculations to find the inradius of each of the other two triangles. However, we can avoid that simply by using the ratios of similarity of the three triangles. The ratio of similarity of triangle BPC to triangle PAB is sqrt(5):1; the ratio of similarity of triangle CDP to triangle PAB is 2:1.

So the sum of the radii of the three circles is
%283-sqrt%285%29%29%2A%281%2Bsqrt%285%29%2B2%29
or
%283-sqrt%285%29%29%2A%283%2Bsqrt%285%29%29+=+9-5+=+4

Such a complicated looking problem and it turns out to have such a "nice" simple answer....