# 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.

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