If the legs of a right-angled triangle are 1: 3, and the hypotenuse is 40, then the length of the height

If the legs of a right-angled triangle are 1: 3, and the hypotenuse is 40, then the length of the height lowered by the hypotenuse is.

First, let’s find the legs of this right-angled triangle.

Let x denote the length of the smaller leg.

According to the condition of the problem, the lengths of the legs of a given right-angled triangle are 1: 3, therefore, the length of the larger leg should be 3x.

Since the hypotenuse of this right-angled triangle is 40, applying the Pythagorean theorem, we can compose the following equation:

x ^ 2 + (3x) ^ 2 = 40 ^ 2,

solving which, we get:

x ^ 2 + 9x ^ 2 = 1600;

10x ^ 2 = 1600;

x ^ 2 = 1600/10;

x ^ 2 = 160;

x = √160 = 4√10.

We find the second leg:

3x = 3 * 4√10 = 12√10.

Find the area S of this triangle:

S = x * 3x / 2 = 4√10 * 12√10 / 2 = 48 * 10/2 = 48 * 5 = 240.

Using the formula for the area of ​​a triangle in terms of the length of the hypotenuse and the height h dropped on this hypotenuse, we find h:

h = 2 * 240/40 = 2 * 6 = 12.

Answer: 12.