+0  
 
0
60
1
avatar+148 

In the diagram, four circles of radius 1 with centres $P$, $Q$, $R$, and $S$ are tangent to one another and to the sides of $\triangle ABC$, as shown.

What is the degree measure of the smallest angle in triangle $PQS$?

AdminMod2  Sep 4, 2017
Sort: 

1+0 Answers

 #1
avatar+76198 
+3

 

Triangle PRS is equilateral.....so angle PSR  = 60° = angle PSQ 

 

And PS  = 2   and  QS  = 4

 

And by the Law of Cosines we have that

 

PQ  =  sqrt ( 2*2 + 4^2  - 2(4)(2) cos 60 )  =  sqrt ( 20 - 8)  = sqrt (12) = 2sqrt(3)

 

And by the Law of Sines, we have

 

sin PSQ / PQ  =  sin PQS / PS

 

sin 60 / 2sqrt (3)   = sin PQS / 2

 

(sqrt (3) / 2) / [ 2sqrt (3) ]  =  sin PQS / 2

 

(1/4)  =  sin PQS / 2

 

sin PQS  =  1/2  ...    thus PQS  = 30°

 

So..since PS is the shortest side in triangle PQS, then angle PQS is the smallest angle   = 30°

 

 

cool cool cool

CPhill  Sep 4, 2017

22 Online Users

avatar
avatar
avatar
avatar
avatar
We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details