Answer:
Step-by-step explanation:
In this geometry, there are three "geometric mean" relations that apply. They can be summarized by the rule ...
a segment touching the hypotenuse is equal to the geometric mean of the segments of the hypotenuse it touches.
For x and y, the two segments will be the short one (4 or 6, respectively) and the entire hypotenuse (4+6=10).
Using this rule, we have ...
x = √(4·10) ≈ 6.3
y = √(6·10) ≈ 7.7
z = √(4·6) ≈ 4.9
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The geometric mean of two numbers is the square root of their product.
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Additional comment
These relations come about because all of the right triangles in this figure are similar. That means corresponding sides are proportional.
x/10 = 4/x ⇒ x² = 40
y/10 = 6/y ⇒ y² = 60
z/6 = 4/z ⇒ z² = 24
