The legs of a right-angled triangle are 15 and 20. Find the height drawn to the hypotenuse.

a = 15, b = 20 – legs, c – hypotenuse, h – height drawn to the hypotenuse.
The problem can be solved in two ways.
1 way
1. By the Pythagorean theorem, we find the length of the hypotenuse:
c = √ (a ^ 2 + b ^ 2);
c = √ (15 ^ 2 + 20 ^ 2) = √ (225 + 400) = √625 = 25.
2. Find the area of ​​a triangle as half the product of its legs:
S = ab / 2;
S = 15 * 20/2 = 300 * 2 = 150.
3. Also, the area of ​​a triangle can be found as half the product of its side and the length of the height drawn to this side. h is the height drawn to the hypotenuse, then:
S = ch / 2.
Substitute the known values:
150 = 25h / 2;
25h = 2 * 150 (proportional);
25h = 300;
h = 300/25 (proportional);
h = 12.
Answer: h = 12.
2 way
In a right-angled triangle, the height that is drawn to the hypotenuse is related to the sides of this triangle by the ratio:
h = ab / c.
By condition, a = 15 conventional units and b = 20 conventional units.
By the Pythagorean theorem, the hypotenuse is equal to:
c = √ (a ^ 2 + b ^ 2);
c = √ (15 ^ 2 + 20 ^ 2) = √ (225 + 400) = √625 = 25.
Substitute the known data into the height formula:
h = 15 * 20/25 = 300/25 = 12.
Answer: h = 12 conventional units.