wow. how do i explain this--Circles and Lenghts of Segments

[itchycontrolyou]

I am asked on my homework to find the radius of each circle. I am going to try and explain this the best I can with words....

I have a circle with a diameter....then a segment in the circle [to the left of the middle pt of the circle.]
**this segment makes a right angle with the diameter segment.
**half of that segment is ten, the bottom half [where the diamter splits it.] is 10.
**now, i am not given the diameter length...but where the segment cuts through the diameter, the segment that it makes, to the left =8.
**how do i find the rest of the diameter?
**teacher gave us the answer---41/4

Good enough description??

 

[galactus]

The distance from the radius through the segment to the perimeter of the circle would be r-8. You have the length of the 'half-chord'.
Use these and Pythagoras to find r(radius), and thus the diameter.

\(\displaystyle \sqrt{(r-8)^{2}+10^{2}}=r\)

The answer you have is for the radius though, not the diameter. The diameter would be twice that.

 

[soroban]

Re: wow. how do i explain this--Circles and Lenghts of Segme

Hello, itchycontrolyou!

If I interpret the problem correctly (and I believe Galactus did, too),
\(\displaystyle \;\;\)the figure looks like this:

              * * *
          *           *
        * : \           *
       *  :  \r          *
        10:   \
      *   :    \          *
      *---+-----* - - - - *
      * 8 :r-8 /          *
          :   /
       *  :  /           *
        * : /           *
          *           *
              * * *

From the right triangle: \(\displaystyle \,(r\,-\,8)^2\,+\,10^2\;=\;r^2\)

Simplify: \(\displaystyle \,r^2\,-\,16r\,+\,64\,+\,100\;=\;r^2\;\;\Rightarrow\;\;16r\,=\,164\;\;\Rightarrow\;\;r\,=\,\frac{41}{4}\)

This is the radius of the circle . . . I hope your teacher knows that.

 

[Denis]

Boy oh boy, Soroban: what a nice circle thou draweth