Geometry probability

[jazziza87]

I have a probability problem that states in circle V, point C is randomly located so that it does not coincide with points R and S. If m. arc RS=one hundred and forty, what is the probability that m.angle RCS=70?And it tells me to give my answer as a fraction. Can someone please tell me what method I need to use in order to work it out. Thanks in advance! :lol:

 

[tkhunny]

Are R, C, and S on the circle?
Do you mean 140º?
My initial impression is Pr(70º) = 0, just because it appears to be a continuous distribution.
Otherwise, "randomly located" is an inadequate description. Uniformly?

The method I would suggest it to retrieve a new problem statement. This one is a bit rough.

 

[jazziza87]

yes, R, C, and S are on the circle and I meant degrees sorry. I don't understand what you are saying though when you say 70 degrees =0. Could you explain a bit more. Thanks.

 

[JakeD]

Look at formula 2 here on the inscribed angle. For what points B does it hold?

 

[soroban]

Hello, jazziza87!

I think I know what you're saying . . .

On a circle, point \(\displaystyle C\) is randomly located so that it does not coincide with points \(\displaystyle R\) and \(\displaystyle S.\)
If \(\displaystyle m(\text{arc}\,RS) = 140^o\), what is the probability that \(\displaystyle \angle RCS\,=\,70^o\)?

Did you make a sketch?

                R
              * * *
          *     |     *
        *       |       *
       *        |        *
                |
      *         |         *
      *         * 140°    *
      *     220° \        *
                  \
       *           \     *
      C o           \   * 
          *           * S
              * * *

Draw chords \(\displaystyle CR\) and \(\displaystyle CS\)

Since arc \(\displaystyle RS\,=\,140^o\), point \(\displaystyle C\) can be anywhere on the major arc \(\displaystyle RCS \,=\,220^o\).
\(\displaystyle \;\;\)Since \(\displaystyle \angle RCS\) is an inscribed angle, it will equal \(\displaystyle 70^o\)

Therefore, the probabiity is the ratio: \(\displaystyle \,\frac{220^o}{360^o}\:=\:\frac{11}{18}\)


Edit: JakeD is too fast for me . . .

 

[tkhunny]

OK, I'll give you the >0, but I still want to know more about the random distribution. Why are we assuming "uniform"?