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Triangle ABC is an isosceles right triangle with AC=4 square root 3 cm. F is the midpoint of hypotenuse AC,

and triange DEF is equilateral. Find the perimeter of triangle DEF.

Guest Jan 9, 2018
 #1
avatar+7336 
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I think we need more information. Where is point  D  or point  E ?

So far, this is all we know....

 

hectictar  Jan 9, 2018
 #2
avatar+91099 
+2

 

 

Using hectictar's diagram..draw altitude BF ......I believe that we have this  :

 

Angle  ABF  =  45°...so angle BAF  = 45°

And  (1/2)AC  =  2√3

 

So...using symmetry.....angle DFA  60°

So  angle FDA  =  75°

 

So.....using the Law of Sines, we can find DF  as follows :

 

(1/2)AC / sin (75)  =  DF / sin (45)

2√3 / sin (75) = DF / sin (45)

2√3 * sin 45 / sin 75  = DF

2√3 *  √2 / 2 / sin (75)  = DF

√6 / sin (75)  = DF  =  

 

√6 /  [ ( 1 + √3 ) / 2√2 ]  =

 

2√12 / [ 1 + √3 ]  =

 

4√3  ( 1 - √3)  / -2  =

 

2√3 ( √3 - 1)  =

 

6 - 2√3  =  DF  ≈ 2.5358

 

And since DEF is equilateral,   the perimeter is 3 times this  =

 

18 - 6√3    units

 

Here's a pic  :

 

 

cool cool cool

CPhill  Jan 10, 2018

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