One of the legs of a right-angled triangle is 89 cm larger than the other and 9 cm less than the hypotenuse. Find the sides of the triangle.
1. Let’s denote the sides of the triangle by a, b and c. Here a and b are legs, c is hypotenuse.
2. By the Pythagorean theorem: c ^ 2 = a ^ 2 + b ^ 2. This is the first equation.
3. Let us compose two more equations according to the problem statement:
a = b + 89; Or b = a – 89;
a = c – 9; Or c = a + 9.
4. Substitute expressions for b and c from the second and third equations into the first. We get the quadratic equation: a ^ 2 – 196 * a + 7840 = 0.
5. The discriminant of the equation D ^ 2 = 38416 – 31360 = 7056. That is, D = 84.
6. Roots of the equation: a = 140 and a = 56. The root a = 56 does not correspond to the conditions of the problem, since in this case b = a – 89 = – 33 <0. The leg cannot have length less than 0.
7. We get the following solution: a = 140, b = 140 – 89 = 51, c = 140 + 9 = 149.
Answer: triangle legs: 140 cm and 51 cm, hypotenuse: 149 cm.
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