Answer:
z = √30
Explanation:
Given similar right triangles with hypotenuse segments marked 3 and 7, you want to know the length of short leg z adjacent to the segment marked 3.
Similar triangles
The fact that all of these triangles are similar gives rise to some "geometric mean" relationships:
hypotenuse/short side = 10/z = z/3
30 = z² . . . . . . . multiply by 3z
z = √30 . . . . . . take the square root
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Additional comment
The other two geometric mean relations are ...
x = √(7·10) . . . . . hypotenuse/long side: 10/x=x/7
y = √(7·3) . . . . . . short side/long side: y/7=3/y
The triangles are similar by AA: each has a right angle, and an angle in common with the largest triangle. Triangles similar to the same triangle are similar to each other.