# The hypotenuse of a right-angled triangle is 13 m and the area is 30 m ^ 2. Find the perimeter of the triangle.

Let’s denote the legs of a right-angled triangle by x and y.

The area of a right-angled triangle by condition is equal to 30 m2 and is found by the formula:

S = ½ x * y

½ x * y = 30

x * y = 60

By the Pythagorean theorem, we write:

x² + y² = 13²

Let’s solve the system of two equations:

x * y = 60

x² + y² = 169

x * y = 60

x² + y² + 2 * x * y = 169 + 2 * x * y

x * y = 60

(x + y) ² = 169 + 2 * 60

x * y = 60

(x + y) ² = 289

x * y = 60

x + y = 17

Let us express y from the second equation and substitute it into the first equation.

y = 17 – x

x * (17 – x) = 60

17x – x² – 60 = 0

x² – 17x + 60 = 0

D = b² – 4ac = 49

x1 = 5, x2 = 12

y1 = 17 – 5 = 12

y 2 = 17 – 12 = 5

Find the perimeter of the triangle:

P = 13 + 12 + 5 = 30 m.

Answer: The perimeter of the triangle is 30 meters.