The sum of the legs of a right-angled triangle is 119 m, and its hypotenuse is 89 m. Find the area of the triangle

The sum of the legs of a right-angled triangle is 119 m, and its hypotenuse is 89 m. Find the area of the triangle using: a) equations b) system of equations c) shortened multiplication formulas.

a) Let one of the legs be equal to x, then the other is equal to 119 – x. By the Pythagorean theorem:

x ^ 2 + (119 – x) ^ 2 = 89 ^ 2;
x ^ 2 + 14161 – 238x + x ^ 2 = 7921;
2x ^ 2 + 14161 – 238x – 7921 = 0;
2x ^ 2 – 238x + 6240 = 0;
x ^ 2 – 119x + 3120 = 0;
D = 119 ^ 2 – 4 * 3120 = 14161 – 12480 = 1681 = 41;
x = (119 ± 41) / 2;
1) x = (119 – 41) / 2 = 78/2 = 39 (m);

119 – x = 119 – 39 = 80 (m);

2) x = (119 + 41) / 2 = 160/2 = 80 (m);

119 – x = 119 – 80 = 39 (m).

b and c) Using the system of equations and formulas for reduced multiplication:

{x + y = 119;
{x ^ 2 + y ^ 2 = 89 ^ 2;
{x + y = 119;
{(x + y) ^ 2 – 2xy = 7921;
{x + y = 119;
{119 ^ 2 – 2xy = 89 ^ 2;
{x + y = 119;
{2xy = 119 ^ 2 – 89 ^ 2;
{x + y = 119;
{2xy = (119 + 89) (119 – 89);
{x + y = 119;
{2xy = 208 * 30;
{x + y = 119;
{xy = 3120.
x and y are the roots of the equation:

t ^ 2 – 119t + 3120 = 0;
t1 = 39; t2 = 80.
Area of ​​a triangle:

S = 1/2 * 39 * 80 = 40 * 39 = 1560 (m ^ 2).

Answer: 1560 m ^ 2.