If the hypotenuse of a right-angled triangle is 25, and the sum of the legs is 31, then the area of the triangle.

Given:
Right angle triangle abc.
c = 25
a + b = 31

To find:
S of triangle abc.

Decision:
Formula for the area of a right triangle:
S = (1/2) * a * b
It is known that
c ^ 2 = a ^ 2 + b ^ 2.
If a + b = 31, then b = 31-a.
Hence:
25 ^ 2 = a ^ 2 + (31-a) ^ 2.
a ^ 2 + 31 ^ 2-62a + a ^ 2 = 25 ^ 2.
2a ^ 2 + 961-62a = 625.
2a ^ 2-62a + 961-625 = 0
2a ^ 2-62a + 336 = 0
a ^ 2-31a + 168 = 0
D = (- 31) ^ 2-4 * 1 * 168 = 961-672 = 289.
a1 = (31-17) / 2 = 7
a2 = (31 + 17) / 2 = 24.
b1 = 31-7 = 24
b2 = 31-24 = 7.
Next, we find the area of the triangle:
S = (1/2) * a * b = 24 * 7/2 = 84.
Answer: the area of the triangle is 84 cm2