Triangle ABC isosceles AB base equal to the square root of 2 angle A = 30 °. find the perimeter.

draw from point C the height to the base AB (CH) the height divides the base in half (AH = HB = √2 / 2) consider the triangle ACH-rectangular according to the theorem of an angle of 30 degrees we find:
let the leg be x, then the hypotenuse is 2x
by the Pythagorean theorem, we find the leg and the hypotenuse

4x ^ 2 = x ^ 2 + 2/4

3x ^ 2 = 1/2 | : 3

x ^ 2 = 1/6

x = 1 / √6 is a leg, and we need a hypotenuse

AC = 2 / √6

P = AC + BC + AB = 2 / √6 + 2 / √6 + √2 = 2 * (2 / √6) + √2 = 4 / √6 + √2 (multiply everything by √6 to get rid of root in the denominator) = 4+ √12 = 4 + 2√3

ANSWER: the perimeter is 4 + 2√3