In a right-angled triangle ABC, the angle is C = 90 degrees, AC = 3, CB = 4. Find the height drawn into the hypotenuse

Given: right-angled triangle ABC;
angle C = 90;
CA = 3;
CB = 4;
CH – height.
Find: CH -?
Decision:
1) consider a right-angled triangle ABC. Then, by the Pythagorean theorem:
AC ^ 2 + CB ^ 2 = AB ^ 2;
3 ^ 2 + 4 ^ 2 = AB ^ 2;
9 + 16 = AB ^ 2;
25 = AB ^ 2;
AB = 5;
2) In a right-angled triangle, each leg is the average proportional between the hypotenuse and the projection of this leg onto the hypotenuse. Then
BC = √ (AB * HB);
4 = √ (5 * HB) (square the right and left parts);
16 = 5 * HB;
HB = 16/5;
HB = 3.2;
3) AC = √ (AB * HA);
3 = √ (5 * HA) (square the right and left parts);
9 = 5 * HA;
HA = 9/5;
HA = 1.8;
4) CH = √AH * HB;
CH = √1.8 * 3.2;
CH = √5.76;
CH = 2.4.
Answer: 2.4.