Find the lengths of the sides of a right-angled equilateral triangle if its perimeter is 21 cm and its area is 36.

The assignment deals with an isosceles right-angled triangle. As you know, this right-angled triangle has equal legs. Therefore, it is necessary to find two unknowns: a – legs and c – hypotenuse.
Further, there is a redundancy of information for solving the triangle. Determine the required values ​​using the known perimeter and area separately, and then compare the results.
Let’s start with the fact that the area of ​​this triangle is 36 cm2. As you know, the area of ​​a right-angled triangle is half the product of the legs. We have: ½ * a * a = 36 cm2 or a2 = 72 cm2. Taking into account that the side of the triangle cannot be negative, from the two roots of the last equation we choose the root: a = 6√ (2) cm.Then, according to the Pythagorean theorem, c ^ 2 = a ^ 2 + a ^ 2 = 2 * a ^ 2 = 2 * 72 cm2 = 144 cm2. We have c = 12 cm.
Now we find the same values ​​using the given perimeter (the sum of all sides) of the triangle (21 cm). We have: 2 * a + c = 21 cm, whence c = (21 – 2 * a) cm.By the Pythagorean theorem, c ^ 2 = 2 * a ^ 2. Therefore, (21 – 2 * a) ^ 2 = 2 * a ^ 2, whence we get the quadratic equation 2 * a ^ 2 – 84 * a + 441 = 0, which has two roots: a1 = (42 – 21√ (2 )) / 2 and a ^ 2 = (42 + 21√ (2)) / 2 (this is a side root, since (42 + 21√ (2)) / 2 = 21 + (21√ (2)) / 2 > 21; the side of a triangle cannot be larger than its perimeter).
Let’s compare the two results. It turns out 6√ (2) ≠ (42 – 21√ (2)) / 2. Let’s stop the calculations and conclude: the redundancy of information in the task led to two unequal values ​​of the leg of the same right-angled triangle!
Conclusion: The given data contradict each other, therefore, it is impossible to unambiguously determine the sides of the triangle.