Question

What is the value of y?

Answer 1

Start with the lower triangle.
It has a right angle and a 45-deg angle.
The third angle is also a 45-deg angle.
The lower triangle is a 45-45-90 triangle.
In a 45-45-90 triangle, the legs are congruent.
The hypotenuse is sqrt(2) times the length of either leg.
The legs are the hypotenuse divided by sqrt(2)
Since the hypotenuse of the lower triangle is 2sqrt(2),
the legs are 2sqrt(2)/sqrt(2) = 2.
Both legs measure 2.

One leg of the lower triangle is the shorter leg of the upper triangle.
The upper triangle is a 30-60-90 triangle.
The longer leg of a 30-60-90 triangle is sqrt(3) times the length
of the shorter leg.
Since the shorter leg measures 2, the longer leg measures 2sqrt(3).

y = 2sqrt(3)