The perimeter of a right-angled triangle is 90 cm and its hypotenuse is 41, find the area of this triangle.

1. Let one leg of a right-angled triangle be x cm, and the second leg y cm. From the condition, the sum of the sides of a right-angled triangle, that is, its perimeter is 90 cm. And the hypotenuse is 41 cm. Let’s make an equation and express one side:
P = x + y + z;
x + y + 41 = 90;
y = 90 – 41 – x;
y = 49 – x;
2. Let’s use the Pythagorean theorem and compose the equation:
z² = x² + y²;
(x² + (49 – x) ²) = 41²;
(x² + (49 – x) ²) = 1681;
x² + 2401 – 98x + x² = 1681;
2x² – 98x + 2401 – 1681 = 0;
2x² – 98x + 720 = 0;
x² – 49x + 360 = 0;
Find the roots by solving the quadratic equation:
Let’s calculate the discriminant:
D = b² – 4ac = (- 49) ² – 4 * 1 * 360 = 2401 – 1440 = 961;
D ›0 means:
x1 = (- b – √D) / 2a = (49 – √961) / 2 * 1 = (49 – 31) / 2 = 18/2 = 9;
x2 = (- b + √D) / 2a = (49 + √961) / 2 * 1 = (49 + 31) / 2 = 80/2 = 40;
Let’s find the second side:
y = 49 – x
If x1 = 9 cm, then y1 = 49 – 9 = 40 cm;
If x2 = 40 cm, then y2 = 49 – 40 = 9 cm;
3. Find the area of ​​a right-angled triangle:
S = 1 / 2xy;
S = 1/2 * 40 * 9 = 180 cm²;
Answer: the area of ​​a right-angled triangle is 180 cm².