In a right-angled triangle, one of the legs is 2 cm less than the hypotenuse – 4 cm less than the hypotenuse.

| April 29, 2021 | education

In a right-angled triangle, one of the legs is 2 cm less than the hypotenuse – 4 cm less than the hypotenuse. Find the sides of this triangle and its area.

Let the length of the hypotenuse AC = X cm, then, by condition, the leg BC = (X – 4) cm, and the leg AB = (X – 2) cm.

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

X ^ 2 = (X – 2) ^ 2 + (X – 4) ^ 2.

X ^ 2 = X ^ 2 – 4 * X + 4 + X ^ 2 – 8 * X + 16.

X ^ 2 – 12 * X + 20 = 0.

Let’s solve the quadratic equation.

D = b ^ 2 – 4 * a * c = (-12) ^ 2 – 4 * 1 * 20 = 144 – 80 = 64.

X1 = (12 – √64) / (2 * 1) = (12 – 8) / 2 = 4/2 = 2. (Not suitable, since then the legs will be equal to 0 and – 2).

X2 = (12 + √64) / (2 * 1) = (12 + 8) / 2 = 20/2 = 10.

AC = 10 cm.

Then BC = 10 – 4 = 6 cm, AB = 10 – 2 = 8 cm.

Determine the area of the triangle.

Savs = AB * BC / 2 = 8 * 6/2 = 24 cm2.

Answer: The sides of the triangle are 6 cm, 8 cm, 10 cm, the area is 24 cm2.



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