In a right-angled triangle, the hypotenuse is 10 cm, the height drawn from the vertex

In a right-angled triangle, the hypotenuse is 10 cm, the height drawn from the vertex of the right angle is 3 cm. Find the segments into which the hypotenuse divides the height.

In a right-angled triangle, the height drawn from the top of the right angle is the average proportional to the two hypotenuse segments formed by it, that is:
h ^ 2 = de,
where h is the height, d and e are the segments into which the height divides the hypotenuse.
Then:
de = 3 ^ 2;
de = 9.
It is also known that:
d + e = 10.
Let’s solve the system of linear equations:
d + e = 10;
de = 9.
In the first equation, we express e in terms of d:
e = 10 – d.
We substitute the resulting expression into the second equation of the system:
d (10 – d) = 9;
10d – d ^ 2 – 9 = 0;
d ^ 2 – 10d + 9 = 0.
Discriminant:
D = 100 – 36 = 64.
d1 = (10 + 8) / 2 = 18/2 = 9.
d2 = (10 – 8) / 2 = 2/2 = 1.
Then:
e1 = 10 – d1 = 10 – 9 = 1.
e2 = 10 – d2 = 10 – 1 = 9.
Answer: the height divides the hypotenuse into segments 9 cm and 1 cm long.