The sum of the legs of a right-angled triangle is 17, and its hypotenuse is 13. Find the area of the triangle.

1. A, B, C – the vertices of the triangle. The angle is straight.

2. We take as x the length of the leg AB, the length of the leg AC (17 – x).

3. Let’s compose the equation:

(17 – x) ² + x² = 13²;

289 – 34x + x² + x² = 169;

2x² – 34x + 120 = 0;

x² – 17x + 60 = 0;

4. An equation can have two roots:

The first value x = (17 + √289 -240) / 2 = (17 + 7) / 2 = 12.

The second value is x = (17 – 7) / 2 = 5.

5. AB = 12 cm or AB = 5 cm.

AC = 17 – 12 = 5 cm or AC = 17 – 5 = 12 cm.

6. The area of the triangle ABC = AB x AC / 2 = 12 x 5/2 = 30 cm².