The hypotenuse of the rectangle of the triangle is 10 cm. And the sum of the legs is 14 cm. Find the area of the triangle.

1. Let’s denote the legs of a right-angled triangle by a and b, and compose equations using the Pythagorean theorem:

{a ^ 2 + b ^ 2 = 10 ^ 2;
{a + b = 14;
{(a + b) 2 – 2ab = 100;
{a + b = 14;
{14 ^ 2 – 2ab = 100;
{a + b = 14;
{196 – 2ab = 100;
{a + b = 14;
{2ab = 196 – 100;
{a + b = 14;
{2ab = 96;
{a + b = 14;
{ab = 48;
{a + b = 14.
2. By the converse Vieta theorem, a and b are the roots of the equation:

p ^ 2 – 14p + 48 = 0;
D / 4 = 7 ^ 2 – 48 = 1;
p = 7 ± 1;
a) p = 7 – 1 = 6 (cm);
b) p = 7 + 1 = 8 (cm).
3. Area of a triangle:

S = 1/2 * ab = 1/2 * 6 * 8 = 24 (cm ^ 2).

Answer: 24 cm ^ 2.