15 points

what is the value of x?

enter your answer, as an exact value, in the box.

X = 3√2.

The top triangle has one angle marked 30 and one marked 90; this makes the other triangle 180-(30+90) = 180-120 = 60.  

In a 30-60-90 triangle, the sides are given by t, t√3, and 2t; t is across from the 30° angle; t√3 is across from the 60° angle; and 2t is across from the 90° angle.  The side marked 6 is across from the 90° angle, so

2t = 6
t = 3

This means that the side between the triangles is 3.

Since the bottom triangle has one angle marked 45 and another marked 90, the third angle would be 180 - (45+90) = 180 - 135 = 45°.  In a 45-45-90 triangle, the sides are listed as t, t, and t√2; the two sides marked t are across from the 45° angles and the side marked t√2 is across from the 90° angle.  This means t = 3, so x = t√2 = 3√2.