The hypotenuse of a right triangle is 13, and the legs are related as 2/3. Find the area of the triangle

Let’s denote by x half the length of the smaller leg of this right-angled triangle.

Then the length of the smaller leg of this triangle is 2x.

From the condition of the problem it is known that the lengths of the legs of this triangle are related as 2: 3, therefore, the length of the larger leg of this triangle is 3x, and the area of ​​this triangle will be 2x * 3x / 2 = 3x ^ 2.

It is also known that the hypotenuse of this triangle is 13, therefore, using the Pythagorean theorem, we obtain the following equation:

(2x) ^ 2 + (3x) ^ 2 = 13 ^ 2.

We solve the resulting equation:

4x ^ 2 + 9x ^ 2 = 13 ^ 2;

13x ^ 2 = 13 ^ 2;

x ^ 2 = 13 ^ 2/13;

x ^ 2 = 13;

x = √13.

Find the area of ​​this triangle:

3x ^ 2 = 3 (√13) ^ 2 = 3 * 13 = 26.

Answer: The area of ​​this triangle is 26.