SOLUTION: ΔABC is an isosceles right triangle with AC=4√3. F is the mid-point of hypotenuse AC, and ΔDEF is equilateral. Find the perimeter of ΔDEF. A) 6(3 - √3) B) 6(2 - √3)

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Question 1189025: ΔABC is an isosceles right triangle with AC=4√3. F is the mid-point of hypotenuse AC, and ΔDEF is equilateral. Find the perimeter of ΔDEF.
A) 6(3 - √3) B) 6(2 - √3) C) 6(5 - 2√3) D) 3(3 - √3) E) 4(5 - 2√3)
https://ibb.co/Vgvz8RG
Found 2 solutions by greenestamps, Edwin McCravy:
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!




Given:
AC=4*sqrt(3)
F is the midpoint of AC, so AF=FC=2*sqrt(3)
DEF is equilateral

To find: The perimeter of DEF

Draw BF intersecting DE at G:



BF bisects DE; and the length of BF is 2*sqrt(3) -- same as AF and FC.

BF divides DEF into two 30-60-90 right triangles.

Let x be the length of EG; then the side length of DEF is 2x, and FG is x*sqrt(3).

BGE is an isosceles right triangle, so the length of BG is also x.

Now we have BF = BG+GF:







Finally, the perimeter of DEF is

ANSWER: A


Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!
I'll pirate Greenestamps picture





By the law of sines,







We know sin(45o), we must find sin(75o).














Rationalize the denominator:















Since ΔDEF is equilateral, its perimeter is 3 times its side EF.

Perimeter of ΔDEF = , choice A)

Edwin
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