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hi!! this is the problem i cant seem to get through today:

 

In triangle $ABC,$ $\angle BCA = \frac{\pi}{6}, AB = 4$ and $AC = 10$.

 

Find the sum of all distinct possible values of $BC.$ If there are no possible values, enter $0$.

 

ive done th eprevious like four parts of the problem but this part is really confusing to me and i just keep getting it wrong. thank you sooo so much in advance for the help!!! :) <3

 Sep 6, 2023
 #1
avatar+14832 
+1

Hello heartSTORM907!

 

There is no result for the distance BC.

\(\overline{BC}\in \{\}\\ \sum(\ \overline{BC}\ )=0\ as\ specified\ in\ the\ task.\)

laugh  !.

 Sep 6, 2023
 #2
avatar+125 
+1

OH MY GOD i almost completely forgot about that part.... i kept thinking i was doing something wrong lol

thank you so so so much!!!!

 Sep 6, 2023
 #3
avatar+126978 
+1

angle BCA =  30°

 

Using the Law of Sines

 

sin BCA / AB  = sin ABC / 10

 

sin (30°) / 4 =  sin ABC /10

 

(1/2) / 4  = sin ABC /10

 

(1/8)  = sin ABC  /10

 

(10/8) = sin ABC

 

(5/4)  = sin ABC        { impossible}

 

cool cool cool

 Sep 6, 2023
 #4
avatar+125 
+1

thank you!!! that was really helpful!

heartSTORM907  Sep 6, 2023

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