The area of a right-angled triangle with a hypotenuse of 10 cm is 24 cm ^ 2, find the larger leg.

Let x and y be the legs of the triangle. By the Pythagorean theorem for the hypotenuse we have:
10 ^ 2 = x ^ 2 + y ^ 2;
Formula for the area of right-angled triangles:
S = 1/2 * x * y; S = 24;
substitute: 24 = 1/2 * x * y,
we find x = 48 / y;
we substitute in the formula for the hypotenuse:
(48 / y) ^ 2 + y ^ 2 = 10 ^ 2;
2304 / y ^ 2 + y ^ 2 = 100; multiply each term on the left and right sides of the equation by y ^ 2;
2304 + y ^ 4 = 100y ^ 2; we have a biquadratic equation;
we introduce an additional variable and replace y ^ 2 = t;
t ^ 2-100t + 2304 = 0;
by Vieta’s theorem, we find the roots:
t1 = 64, t2 = 36;
we find:
y ^ 2 = 64; y1 = √64 = 8; do not take negative values.
y ^ 2 = 36; y2 = √36 = 6;
x1 = 48 / y1 = 48/8 = 6;
x2 = 48 / y2 = 48/6 = 8;
The larger leg is 8 cm.