+0  
 
0
143
9
avatar

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

 Feb 9, 2021
 #3
avatar
0

Angle AYX = 120 degrees

 Feb 9, 2021
 #5
avatar+113779 
+3

Here is the diagram, and I have introduced a new point, E

 

BX and BY are both tangents to the circle, Hence they are the same length.

So  triangle EYB and triangle EXB are congruent.

So angle YEB = 90 degrees.

So angle EYB = 180-90-30 = 60 degrees

So angle AYX = the supplement of 60 degrees,  180-60 = 120 degrees

 

 

 Feb 9, 2021
 #7
avatar+120287 
+1

Very nice, Melody   !!!!

 

 

cool cool cool

CPhill  Feb 9, 2021
 #9
avatar+113779 
0

Thanks Chris.  laugh

Melody  Feb 9, 2021
 #6
avatar
+1

Thank you Melody! I got 120 later from old posts other people got, but not quite the same question. Your explanation is brilliant

 Feb 9, 2021
 #8
avatar+113779 
0

You are welcome.

I should have seen the answer straight away but I didn't.   

Maybe I will next time....    

 

anyway, I am glad I could help.

Melody  Feb 9, 2021

20 Online Users

avatar