the larger leg of a right triangle is 7ft longer than its samller leg. the hypotenuse is 1ft longer than the larger leg. how many feet long is the samller leg?

smaller leg ---- x , where x is a positive number

larger leg ------ x+7
hypotenuse ----- x+8

x^2 + (x+7)^2 = (x+8)^2
x^2 + x^2 + 14x + 49 = x^2 + 16x + 64
x^2 - 2x - 15 = 0
(x-5)(x+3) = 0
x = 5 or x is a negative ----> no good

so the 3 sides are:
5 , 12 and 13

check:
is 5^2 + 12^2 = 13^2 , sure is!!!

To find the length of the smaller leg of the right triangle, let's assign variables to the unknowns. Let's say the length of the smaller leg is "x" feet.

According to the problem, the larger leg is 7 feet longer than the smaller leg. So, the length of the larger leg can be expressed as "x + 7" feet.

The hypotenuse is 1 foot longer than the larger leg. Therefore, the length of the hypotenuse is "x + 7 + 1" feet, which simplifies to "x + 8" feet.

Now, we can apply the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, the equation would be:

(x^2) + ((x + 7)^2) = (x + 8)^2

Expanding and simplifying this equation:
x^2 + (x^2 + 14x + 49) = x^2 + 16x + 64

Combine like terms:
2x^2 + 14x + 49 = x^2 + 16x + 64

Rearrange the terms to set the equation to zero:
x^2 + 2x^2 - x^2 + 14x - 16x + 49 - 64 = 0

Simplify further:
x^2 - 2x - 15 = 0

Now, we have a quadratic equation. We can solve it by factoring:

(x - 5)(x + 3) = 0

Setting each factor to zero:
x - 5 = 0 or x + 3 = 0

Solving for x:
x = 5 or x = -3

As we are measuring the length of a side of a triangle, the length cannot be negative. Thus, the length of the smaller leg is 5 feet.

Therefore, the smaller leg of the right triangle is 5 feet long.