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

September 25, 2021 | education

| 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.

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