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In triangle $PQR$, we have $\angle P = 90^\circ$, $QR = 20$, and $\tan R = 4\sin R$. What is $PR$?

 
michaelcai  Nov 14, 2017
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QR is the hypotenuse of this right triangle....and we have that.....

 

tan R  = 4 / sin R

sinR * tanR  = 4

sin^2R / cosR =  4

(1 - cos^2R) / cosR  = 4

1 - cos^2R = 4cosR

cos^2R + 4cosR - 1  = 0

 

Let x = cosR  ........ so....

 

x^2 + 4x - 1   = 0

 

Solving this for x  gives  

 

x = - √5 - 2       or   x  =   √5 - 2

 

However....since R  is acute......the second value will only be good for the cosine

 

So x =  cos R  =  √5 - 2

 

So

 

Cos R  =  PR / QR

 

√5 - 2  =  PR / 20

 

So     ....    PR   =  20 ( √5 - 2) 

 

 

cool cool cool  

 
CPhill  Nov 14, 2017

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