The hypotenuse of the triangle is 17 cm, and the sum of the legs is 23 cm. Find the area of the triangle.

Let the length of the leg AB be equal to X cm, then, by condition, the length of the second leg will be equal to (23 – X) cm.

Then, by the Pythagorean theorem, AC ^ 2 = AB ^ 2 + BC ^ 2.

17 ^ 2 = X ^ 2 + (23 – X) ^ 2.

289 = X ^ 2 + 529 – 46 * X + X ^ 2.

2 * X ^ 2 – 46 * X + 240 = 0.

X ^ 2 – 23 * X + 120 = 0.

Let’s solve the quadratic equation.

D = b ^ 2 – 4 * a * c = (-23) ^ 2 – 4 * 1 * 120 = 529 – 480 = 49.

X1 = (23 – √49) / 2 * 1 = (23 – 7) / 2 = 16/2 = 8.

X2 = (23 + √49) / 2 * 1 = (23 + 7) / 2 = 30/2 = 15.

If AB = 8 cm, then BC = 23 – 8 = 15 cm.

Then Savs = AB * BC / 2 = 8 * 15/2 = 60 cm2.

Answer: The area of the triangle is 60 cm2.