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.
+9465 I think we need more information. Where is point D or point E ?
So far, this is all we know....
+128406
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 :
