SOLUTION: Circles M & N (centered at points M and N respectively) are tangent to each other and to AC & BC. If the radius of the circle with center N is 5/(1+sqrt2) m, what is the radius of

Algebra ->  Circles -> SOLUTION: Circles M & N (centered at points M and N respectively) are tangent to each other and to AC & BC. If the radius of the circle with center N is 5/(1+sqrt2) m, what is the radius of       Log On


   

Question 1151182: Circles M & N (centered at points M and N respectively) are tangent to each other and to AC & BC. If the radius of the circle with center N is 5/(1+sqrt2) m, what is the radius of M, in meters?
Diagram: https://imgur.com/a/MKn3C2J
Found 3 solutions by greenestamps, ikleyn, jim_thompson5910:
Answer by greenestamps(13216) About Me  (Show Source):
You can put this solution on YOUR website!


Circles M & N (centered at points M and N respectively) are tangent to each other and to cross%28AC%29 AB & BC. If the radius of the circle with center N is 5/(1+sqrt2) m, what is the radius of M, in meters?

Let r be the radius of circle N and R be the radius of circle M.

Let D and E be the points of tangency of circles N and M, respectively, with BC; let F be the point of tangency of the two circles.

Triangles BDN and BEM are isosceles right triangles.

MB+=+MF%2BFN%2BNB+=+%28R%29%2B%28r%29%2B%28r%2Asqrt%282%29%29+=+R%2Asqrt%282%29

Then

%28R%29%2B%28r%29%2B%28r%2Asqrt%282%29%29+=+R%2Asqrt%282%29
R%2Br%281%2Bsqrt%282%29%29+=+R%2Asqrt%282%29

Given that r=5%2F%281%2Bsqrt%282%29%29,

R%2B%285%2F%281%2Bsqrt%282%29%29%29%2A%281%2Bsqrt%282%29%29+=+R%2Asqrt%282%29

R%2B5+=+R%2Asqrt%282%29

5+=+R%2Asqrt%282%29-R+=+R%28sqrt%282%29-1%29



ANSWER: The radius of circle M in meters is 5%2A%28sqrt%282%29%2B1%29

Answer by ikleyn(52914) About Me  (Show Source):
You can put this solution on YOUR website!
.

Notice that  NEITHER  wording part of the post,  NOR  its plotting part do not specify the measure of the angle ABC
and do not declare that this angle is the right angle.

Therefore,  this problem,  as it is worded,  printed,  posted and presented,  is    D E F E C T I V E.


/\/\/\/\/\/\/\/

Added after reading the post by Jim. 

    Jim, the first line of your post is


        Angle ABC is 90°.


    Could you please tell me what is a rationale for this your statement ?


    It is not stated in the original post and not declared in the plot accompanying the original post.


    It is why I said that the presented original post is DEFECTIVE.


Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

@greenestamps has a great solution. I don't have much to add other than a diagram that might help supplement his work.



Edit: @ikleyn makes a great point that it is not clearly stated anywhere that ABC is 90 degrees. Hopefully that information is posted elsewhere in your textbook (perhaps in a previous problem). The visual cue that ABC could be 90 degrees came from the fact that ABC looks like a square angle. If ABC is not 90 degrees, then the whole thing falls apart.