There are 3 right triangles in this picture, and they are all mutually similar.
• the smallest triangle has legs 4 and y, and hypotenuse z
• the middle one has legs y and 16, and hypotenuse x
• the largest one has legs z and x, and hypotenuse 16 + 4 = 20
Note the order in which I've listed the sides (shortest to longest). Because these triangles are similar, the corresponding sides occur in a fixed ratio. In particular, we have
4/y = y/16 = z/x … (ratios of each side of the smallest and middle triangles)
4/z = y/x = z/20 … (ratios of the sides of the smallest and largest triangles)
y/z = 16/x = x/20 … (ratios of the sides of the middle and largest triangles)
We just want to find x, which we can do easily with the last equality,
16/x = x/20
Solve for x :
20 • 16/x = 20 • x/20
320/x = x
x • 320/x = x • x
320 = x²
⇒ x = √320 = √(64 • 5) = 8√5
