the hypotenuse of a right triangle is 1 inch longer than one leg and 8 inches longer than the other. Find the length of each side of the triangle.

Let l1, l2 be the legs.

l1^2 + l2^2 = h^2
but h= l1+1
and h= l2 + 8

solve these for l1, l2, put them in the first equation and solve for h.
Then solve for the others.

To find the length of each side of the triangle, let's use the variables l1 and l2 to represent the lengths of the legs, and h to represent the length of the hypotenuse.

According to the problem, the hypotenuse of the right triangle is 1 inch longer than one leg and 8 inches longer than the other.

So, we have the following equations:
h = l1 + 1
h = l2 + 8

Now, we can use the Pythagorean theorem to find another equation. The Pythagorean theorem states that in a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

Therefore, we have:
l1^2 + l2^2 = h^2

Now, let's solve these equations step by step:
1. Substitute the values for h from the second equation into the first equation: l1 + 1 = l2 + 8
Rearrange the equation to get: l1 - l2 = 7 (Equation A)

2. Rewrite the second equation as h^2 - l2^2 = l1^2

3. Substitute the value of h from the first equation into the rewritten second equation:
(l1 + 1)^2 - l2^2 = l1^2
Expand and simplify:
l1^2 + 2l1 + 1 - l2^2 = l1^2
2l1 + 1 - l2^2 = 0
Rearrange the equation to get: 2l1 = l2^2 - 1 (Equation B)

4. Now, we have two equations (Equation A and Equation B) with two variables. We can solve this system of equations to find the values of l1 and l2.

5. Substitute the value of l1 from Equation A into Equation B:
2(l2 - 7) = l2^2 - 1
Expand and simplify:
2l2 - 14 = l2^2 - 1
Rearrange the equation to get: l2^2 - 2l2 - 13 = 0

6. Solve the quadratic equation using factoring, completing the square, or the quadratic formula to find the values of l2.

7. Once you have the value of l2, substitute it back into either Equation A or Equation B to find the corresponding value of l1.

8. Finally, substitute the values of l1 and l2 into the first equation (l1^2 + l2^2 = h^2) to find the length of the hypotenuse, h.

By following these steps, you will be able to find the lengths of each side of the triangle.