Answer:
BD = 5√3
AD = 10
CD = 10√3
Explanation:
You want the missing side lengths in a right-triangle geometry that shows the hypotenuse divided into lengths 5 and 15 by the altitude.
Geometric mean relations
In this geometry, all of the right triangles are similar. The similarity proportions can be solved to give three geometric mean relationships:
BD = √(AB·CB) = √(5·15) = 5√3
AD = √(AB·AC) = √(5·20) = 10
CD = √(CB·CA) = √(15·20) = 10√3
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Additional comment
It can be handy to remember the geometric mean relations. They can be thought of as "each side is equal to the geometric mean of the hypotenuse segments it touches." (Note that touching the end is interpreted as touching the nearest part and the whole.)
For example, the similarity proportion for AD is ...
hypotenuse/(short side)
AD/AB = AC/AD ⇒ AD² = AB·AC ⇒ AD = √(AB·AC)
These are called "geometric mean" relations because the geometric mean of two numbers is the square root of their product.