The flower garden has the shape of a right triangle. 5ft of a perennial border forms the hypotenuse of the triangle, and one leg is 1ft longer than the other leg.Find the lengths of the legs.

a^2+b^2=C^2

x^2+(x+1)^2=5^2
the other legs are 3ft and 4ft

You cut square corners with side lengths that are whole numbers from a piece of cardboard with dimensions 20 inches by 30 inches. You then fold the cardboard to create a box with no lid.

Let's call one leg of the triangle x and the other leg x+1 (since one leg is 1ft longer than the other).

According to the Pythagorean theorem, in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

So, we can write the equation as:

x^2 + (x+1)^2 = 5^2

Expanding and simplifying the equation:

x^2 + x^2 + 2x + 1 = 25

Combining like terms:

2x^2 + 2x + 1 = 25

Subtracting 25 from both sides:

2x^2 + 2x - 24 = 0

Dividing by 2 to simplify the equation:

x^2 + x - 12 = 0

Next, we need to factor the quadratic equation:

(x + 4)(x - 3) = 0

So, we have two possible solutions for x: x = -4 and x = 3.

Since lengths cannot be negative, we discard the negative solution.

Therefore, the length of one leg is x = 3ft.

And the length of the other leg is x+1 = 3+1 = 4ft.

So, the lengths of the legs of the right triangle are 3ft and 4ft.

To solve this problem, we can use the Pythagorean theorem. According to the theorem, in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Let's assume that one leg of the right triangle is x feet long. According to the problem, the other leg is 1 foot longer, so its length will be x + 1.

Now, let's use the Pythagorean theorem to set up an equation:

(x)^2 + (x + 1)^2 = (5)^2

Simplifying the equation:

x^2 + (x^2 + 2x + 1) = 25

Combining like terms:

2x^2 + 2x + 1 = 25

Subtracting 25 from both sides:

2x^2 + 2x - 24 = 0

Dividing the equation through by 2:

x^2 + x - 12 = 0

Now, we need to factor this quadratic equation:

(x + 4)(x - 3) = 0

Setting each factor equal to zero:

x + 4 = 0 or x - 3 = 0

Solving for x:

x = -4 or x = 3

Since the length of a side cannot be negative, we discard the solution x = -4.

Therefore, the length of one leg of the right triangle is x = 3 feet, and the length of the other leg is x + 1 = 3 + 1 = 4 feet.

So, the lengths of the legs are 3 feet and 4 feet.