SOLUTION: The centers of two circles are 51 centimeters apart. The smaller circle has a radius of 7 centimeters and the larger circle has a radius of 9 centimeters. Find the length of the co

Algebra ->  Circles -> SOLUTION: The centers of two circles are 51 centimeters apart. The smaller circle has a radius of 7 centimeters and the larger circle has a radius of 9 centimeters. Find the length of the co      Log On


   

Question 782939: The centers of two circles are 51 centimeters apart. The smaller circle has a radius of 7 centimeters and the larger circle has a radius of 9 centimeters. Find the length of the common internal tangent (from the point of tangency on one circle to the point of tangency on the other circle). Give both an exact answer, and an approximate answer rounded to the nearest ten thousandth.
Answer by Edwin McCravy(20059) About Me  (Show Source):
You can put this solution on YOUR website!

We let AC = x and therefore CD = 51-x.

Triangles ABC and DEC are similar and so their sides are
proportional.

AB%2F%28DE%29 %22%22=%22%22 AC%2FDC

7%2Fx %22%22=%22%22 9%2F%2851-x%29

Cross-multiply:

 9x = 7(51-x)

 9x = 357 - 7x

16x = 357

  x =  = 7² + BC²

127449%2F256 = 49 + BC²

Clear of fractions by multiplying through by 256

127449 = 12544 + 256BC²

114905 = 256BC²

114905%2F256 = BC²

sqrt%28114905%2F256%29 = BC²

sqrt%2849%2A2345%29%2F16%29 = BC

7%2Asqrt%282345%29%2F16%29 = BC

I could use ratio and proportion to get EC, but since
BC came out so complicated, I'll use the Pythagorean theorem
again.

DC = 51-x = 51 - 357%2F16 = 816%2F16-357%2F16 = 459%2F16 

DC² = DE² + EC²

%28459%2F16%29%5E2 = 9² + EC²

210681%2F256 = 81 + EC²

Clear of fractions by multiplying through by 256

210681 = 20736 + 256EC²

189945 = 256EC²

189945%2F256 = EC²

sqrt%28189945%2F256%29 = EC²

sqrt%2881%2A2345%29%2F16%29 = EC

9%2Asqrt%282345%29%2F16%29 = EC

So we find the internal tangent BE,

BE = BC + EC = 7%2Asqrt%282345%29%2F16%29 + 9%2Asqrt%282345%29%2F16%29 = 16%2Asqrt%282345%29%2F16%29 = cross%2816%29%2Asqrt%282345%29%2Fcross%2816%29%29 = √2345

Exact answer = √2345

approximate answer rounded to the nearest ten thousandth = 48.4252

Edwin