Answer:
x = 24
y = 8√10
Explanation:
You want the lengths marked x and y in the right triangle figure shown.
Missing length
The short segment of the large triangle's hypotenuse is 80 -72 = 8 units.
Geometric mean
The triangles in this figure are all similar. The proportions that relate corresponding sides give rise to three "geometric mean" relationships. In effect, the segments marked x and y, and the unmarked segment at bottom left, are the geometric mean of the two hypotenuse segments they touch:
x = √(8·72) = 24
y = √(8·80) = 8√10
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Additional comment
As an example of the proportion and the relation it gives, consider the ratio of the short side to the hypotenuse of the largest and smallest triangles:
y/80 = 8/y
y² = 8·80
y = √(8·80)
The relation for x is based on the ratio of short side to long side in the smaller triangles.