# The legs are related as 1: 3 to find the height drawn from the top of the right angle to the hypotenuse

The legs are related as 1: 3 to find the height drawn from the top of the right angle to the hypotenuse. If hypotenuse = 40.

Let’s find the lengths of the legs of this right-angled triangle.

Let’s denote by x the length of the smaller leg.

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

By the condition of the problem, the hypotenuse of this right-angled triangle is 40.

Using the Pythagorean theorem, we get the following equation:

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

We solve the resulting equation:

x ^ 2 + 9x ^ 2 = 1600;

10x ^ 2 = 1600;

x ^ 2 = 1600/10;

x ^ 2 = 160;

x = √160;

x = 4√10.

We find a larger leg:

3x = 3 * 4√10 = 12√10.

Knowing both legs, we find the area S of this triangle:

S = 4√10 * 12√10 / 2 = 24 * 10 = 240.

Knowing the area of ​​this triangle and the hypotenuse c, we find the height h drawn to the hypotenuse:

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

Answer: the length of the height drawn from the top of the right angle to the hypotenuse is 12.

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