Fast right triangle abc has a right angle c, and the measure of angle a=60 degrees. given ac=4 units, which of the following is the exact length of segment bc? []= square root sign 4 units 4[3] units 4[2] units 8 units

The correct answer is:  [B]:  " 4√3 units " .

Explanation:

Given the following right triangle:  (NOTE:  Not to scale at all!) :

   A
   |   \ 
   |     \
   | 60  \  
   |         \ 
4 |          \ 
   |           \
   |---         \
   |_ |___ _\            
C      x        B


Solve for "x" .

"x" = the exact length of segment "BC" .

Note:  Remember the mnemonic:  "SOH CAH TOA" ; 

in which:  "SOH :  sin = opposite/ hypotenuse;

                 "CAH: cos = adjacent/ hypotenuse; 

                 "TOA":  tan = opposite/adjacent.

With the information we have at this point, we can use:  "TOA";

that is:  "tangent = opposite/ adjacent" ; to solve for "x" ;

that is:  tan 60 = x / 4 ;  Solve for "x" ; 

↔  x / 4 = tan 60 ; 

Multiply EACH SIDE of the equation by "4" ; 
 to isolate "x" on one side of the equation ; & to solve for "x" ; 

   4 * (x/4) = 4 tan 60 ; 

x = 4 * tan 60 ; 

x = 4* 1.732050807569 ; 

x = 6.928203230276 ;

The answer choices given are: 

Choice  [A}:  4 units;  which is incorrect; since "4 ≠ 6.928203230276" ; 

Choice  [B]:  4√3 units; 

Choice  [C]:  4√2 units; 

Choice  [D]:  8 units; which is incorrect; since "8 ≠ 6.928203230276" .

We are left with choices [B] and [C].

Let us try both answer choices:

Choice [B]:  "4*√3" = 4 * 1.7320508075688773 = 6.9282032302755092 ;
       → which is the correct answer ;
                  since:  "6.9282032302755092 = 6.9282032302755092" . 

Choice [C]:  "4*√2" = 4 * 1.414213562373095 = 5.6568542494923802 ;
                            This choice is INCORRECT ;  since:
                        "5.6568542494923802 ≠  6.9282032302755092" .

The correct answer is:  [B]:  " 4√3 units " .

Set up a system of equations