In a right-angled triangle, the length of one leg is equal to the arithmetic mean of the lengths of the second

In a right-angled triangle, the length of one leg is equal to the arithmetic mean of the lengths of the second leg and the hypotenuse. Find the sine of the smaller angle of this triangle if its hypotenuse is 10 cm larger than the second leg.

Let the second leg be x, then the hypotenuse (x + 10). Let’s calculate the first leg (its length is equal to the arithmetic mean of the hypotenuse and the second leg):

(x + (x + 10)): 2 = x + 5.

Let’s compose an equation using the Pythagorean theorem:

(x + 5) ^ 2 + x ^ 2 = (x + 10) ^ 2.

Having calculated that x = 15, we find the first leg and the hypotenuse:

(x + 5) = 20;

(x + 10) = 25.

Since the smaller angle lies against the smaller side, we find the sine of the smaller angle (the ratio of the opposite leg to the hypotenuse):

15: 25 = 0.6.