Melody

17)  Bolean contour mapping to solve probablility questions.     (29/9/17)

Once again it is not displaying properly only this time it is the LaTex that is not displaying correctly  

Jeff123,

It would be really nice if you would comment on my answers. I think my answers are correct but this technique is quite new to me.  If you have any reason to think the answers are incorrect (like maybe the probability answer is printed in your text book) it would be really nice if you share what you know. 

QUESTION 2

(2) Three points are selected randomly on the circumference of a circle. What is the probability that the triangle formed by these three points contains the center of the circle?

I drew this picture in Geogebra. The real thing is interactive and by playing with it I concluded that if all the angles of the triangle are acute then the centre will be included. 

Now it is very similar to the first problem only maybe easier.  

Let the 3 angles be  x, 180-y  and y-x   

Where y>x and each of thes angles is acute.

\(0 x \qquad (3)\\ x<90\qquad (4)\\ 180-y<90\\ -y<-90\\y>90 \qquad (5)\\ y-x<90\\ y

Here is the Boolean contour map  so if y>x then the probability is 1/8

BUT if x>y then you have the inverse of this region. I mean this region reflected over  the line y=x

SO

It seems to me that the probability that the centre will be in the triangle is 1/4

Happy birthday !  

Circle \;\;\Gamma\;\; \text{ is the incircle of }\triangle ABC \text{ and is also the circumcircle of }\triangle XYZ\text{ The point X is on }\overline{BC} \text{ point Y is on }\overline{AB}, \textand the point Z is on }\overline{AC}.\;\; If\;\; $\angle A=40^\circ, \;\;\angle B=60^\circ,\;\; and \;\;\angle C=80^\circ, \text{what is the measure of }\angle AYX\; ?

\(\text{Circle }\Gamma\;\; \text{ is the incircle of } \triangle ABC \text{ and is also the circumcircle of } \triangle XYZ\\ \text{ The point X is on }\overline{BC} \text{and the point Y is on }\overline{AB}, \\ \text{and the point Z is on }\overline{AC}.\\ If\;\; \angle A=40^\circ, \;\;\angle B=60^\circ,\;\; and \;\;\angle C=80^\circ, \text{what is the measure of }\angle AYX\; ? \)

   Let O be the centre of the circle.

  OX=OY                                 equal radio

  So

  OXY is an isosceles triangle

  120+2

  so

  < AYX =

I am sorry, this was a full answer but 3/4 of it has been deleted.

There is obviously a software problem.       

I shall report it as a problem :/

You need to use brackets darkcalculus :)

23+9/7-3=8

should be

(23+9)/ (7-3)=8

Not if you do not ask a question :/

Rotate 90º Counter clockwise Around the Origin And then Reflect Over the X axis With (-8,0),(-11,-5),(-4,-10),(-6,-4)