The area of a right-angled triangle is 65. One of the legs is 3 larger than the other. Find the smaller leg.

Let us denote by a the length of the larger leg of this right-angled triangle.

In the initial data for this task, it is reported that one of the legs of this right-angled triangle is 3 more than the other, therefore, the length of the smaller of the legs of this triangle should be equal to a – 3.

Also in the problem statement it is said that the area of ​​this triangle is 65.

Since the area of ​​any right-angled triangle is half the product of its legs, we can draw up the following equation:

a * (a – 3) / 2 = 65.

solving which, we get:

a * (a – 3) = 2 * 65;

a ^ 3 – 3a = 130;

a ^ 3 – 3a – 130 = 0;

a = (3 ± √ (9 + 4 * 130)) / 2 = (3 ± √529) / 2 = (3 ± 23) / 2;

a1 = (3 + 23) / 2 = 26/2 = 13;

a2 = (3 – 23) / 2 = -20 / 2 = -10.

Since the length of the leg cannot be negative, the value a = -10 is not suitable.

Find the smaller leg:

a – 3 = 13 – 3 = 10.

Answer: 10.