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)

Algebra ->  Triangles -> 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)       Log On


   

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) About Me  (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:

2%2Asqrt%283%29=x%2Bx%2Asqrt%283%29
2%2Asqrt%283%29=x%281%2Bsqrt%283%29%29
x=2%2Asqrt%283%29%2F%281%2Bsqrt%283%29%29
x=%282%2Asqrt%283%29%29%281-sqrt%283%29%29%2F%28%281%2Bsqrt%283%29%29%281-sqrt%283%29%29%29
x=%282%2Asqrt%283%29-6%29%2F%281-3%29+=+%286-2%2Asqrt%283%29%29%2F2+=+3-sqrt%283%29

Finally, the perimeter of DEF is 6x=6%283-sqrt%283%29%29

ANSWER: A 6%283-sqrt%283%29%29

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





By the law of sines,

EF%2Fsin%28C%29%22%22=%22%22FC%2Fsin%28%22%3CFEC%22%29

EF%2Fsin%2845%5Eo%29%22%22=%22%222sqrt%283%29%2Fsin%2875%5Eo%29

EF%2Asin%2875%5Eo%29%22%22=%22%222sqrt%283%29sin%2845%5Eo%29

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

sin%2875%5Eo%29%22%22=%22%22sin%2845%5Eo%2B30%5Eo%29%22%22=%22%22sin%2845%5Eo%29cos%2830%5Eo%29%2Bcos%2845%5Eo%29sin%2830%5Eo%29%22%22=%22%22
%28sqrt%282%29%2F2%29%28sqrt%283%29%2F2%29%2B%28sqrt%282%29%2F2%29%281%2F2%29%22%22=%22%22sqrt%286%29%2F4%2Bsqrt%282%29%2F4%22%22=%22%22%28sqrt%286%29%2Bsqrt%282%29%29%2F4

EF%2A%28%28sqrt%286%29%2Bsqrt%282%29%29%2F4%29%29%22%22=%22%222sqrt%283%29%2A%28sqrt%282%29%2F2%29

EF%2A%28sqrt%286%29%2Bsqrt%282%29%29%2F4%29%29%22%22=%22%222sqrt%283%29%28sqrt%282%29%2F2%29

EF%2A%28%28sqrt%286%29%2Bsqrt%282%29%29%2F4%29%29%22%22=%22%22sqrt%286%29

EF%2A%28%28sqrt%286%29%2Bsqrt%282%29%29%29%29%22%22=%22%224%2Asqrt%286%29

EF%22%22=%22%224sqrt%286%29%2F%28%28sqrt%286%29%2Bsqrt%282%29%29%29%29

Rationalize the denominator:

EF%22%22=%22%22%284sqrt%286%29%29%2F%28+%28sqrt%286%29%2Bsqrt%282%29+%29%29%22%22%2A%22%22%28sqrt%286%29-sqrt%282%29%29%2F%28sqrt%286%29-sqrt%282%29%29

EF%22%22=%22%22%28++4sqrt%286%29%28sqrt%286%29-sqrt%282%29%29%29%2F%286-2%29

EF%22%22=%22%22%284sqrt%286%29%28sqrt%286%29-sqrt%282%29%29%29%2F%284%29

EF%22%22=%22%22sqrt%286%29%28sqrt%286%29-sqrt%282%29%29

EF%22%22=%22%226-sqrt%2812%29

EF%22%22=%22%226-sqrt%284%2A3%29

EF%22%22=%22%226-2sqrt%283%29%22%22=%22%222%283-sqrt%283%29%29

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

Perimeter of ΔDEF = 6%283-sqrt%283%29%29, choice A)

Edwin