Answer:
4√11
Explanation:
You want the length of the altitude BD in right triangle ABC if it divides the hypotenuse into segments AD=44 and DC=4.
Similar triangles
The altitude to the hypotenuse in a right triangle divides the figure into similar triangles:
∆ABC ~ ∆BDC ~ ∆ADB
Proportional sides
The ratios of the leg lengths in these triangles are the same, so we have ...
AD/BD = BD/DC . . . . . . . ratio of short leg to long leg
44/BD = BD/4 . . . . . . . . . with given numbers
BD² = 4·44 = 4²·11 . . . . . multiply by 4·BD
BD = 4√11 . . . . . . . . . . . take the square root
The length of BD is 4√11.
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Additional comment
The altitude to the hypotenuse of a right triangle is the geometric mean of the segment lengths of the hypotenuse, as we see above:
BD = √(AD·DC)
There are two other geometric mean relations in this geometry. Each is a direct result of the proportional relation between corresponding sides.
AB = √(AD·AC)
CB = √(CD·CA)
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