Answer:
x = 3√3
y = 6
Step-by-step explanation:
The geometric mean relations between segments intersecting the long hypotenuse and its parts can be used to find the values of interest.
Altitude
x = √(9·3) = 3√3
Short side
y = √((9+3)·3) = 6
__
Additional comment
These right triangles are all similar, so corresponding sides are proportional. When the proportions are solved for a missing side, a geometric mean relation results. (The geometric mean of 'a' and 'b' is √(ab).)
Identify the segments above the horizontal line as w, x, y. (x and y are already identified in this figure.)
The ratio of short side to long side is ...
x/9 = 3/x ⇒ x² = 9·3 ⇒ x = √(9·3)
The ratio of short side to hypotenuse is ...
y/(9+3) = 3/y ⇒ y² = (9+3)·3 ⇒ y = √((9+3)·3)
Likewise, the ratio of long side to hypotenuse is ...
w/(9+3) = 9/w ⇒ w² = (9+3)·9 w = √((9+3)·9) = 6√3
