because of the rules for 30 60 90 triangles, (opposite side to 30 is x, opposite side to 60 is x root 3, opposite side to 90 is 2x), we know that 8 root 2, which is opposite to the 90 degree angle, is equal to 2x. that means that x = 4 root 2, so the side across from the thirty degree angle is 4 root 2. the smaller triangle is a 45 45 90 triangle, which has a similar rule (side opposite to 45 degree angles is x, side opposite to 90 degree is x root 2.) since we know the side opposite from the 90 degree angle is 4 root 2 (see above), then 4 root 2 = x root 2. so divide both sides by root two to get x = 4.