The longer leg of a right triangle is 3 ft longer than three times the shorter leg. The hypotenuse is 3 ft shorter than four times the shorter leg. Find the lengths of the three sides of the right triangle. (Please provide steps/explanation to solve this problem.)

Let's denote the lengths of the shorter leg, longer leg, and the hypotenuse as x, y, and z, respectively.

According to the given information, the longer leg is 3 ft longer than three times the shorter leg. This can be expressed as:

y = 3x + 3 ----(1)

Additionally, the hypotenuse is 3 ft shorter than four times the shorter leg. Therefore, we can write:

z = 4x - 3 ----(2)

To find the lengths of the three sides, we need to solve these two equations simultaneously.

First, let's solve equation (1) for x:

y = 3x + 3

Subtracting 3 from both sides, we get:

y - 3 = 3x

Now, divide both sides by 3:

(x = (y - 3) / 3) ----(3)

Now, substitute this value of x into equation (2):

z = 4x - 3

Replacing x with its value from equation (3), we have:

z = 4((y - 3) / 3) - 3

Simplifying this expression, we get:

z = (4y - 12) / 3 - 3

Multiplying through by 3 to clear the fraction, we obtain:

3z = 4y - 12 - 9

Combining like terms, we have:

3z = 4y - 21

Now, let's solve this equation for y:

4y = 3z + 21

Dividing through by 4, we get:

y = (3z + 21) / 4 ----(4)

We now have expressions for both x and y in terms of z. To find the lengths of the three sides, substitute these expressions into the Pythagorean theorem:

x^2 + (3x + 3)^2 = ((3z + 21) / 4)^2 + z^2

Simplifying this, we obtain:

x^2 + 9x^2 + 18x + 9 = (9z^2 + 126z + 441) / 16 + z^2

Multiplying through by 16 to clear the fraction, we have:

16x^2 + 144x^2 + 288x + 144 = 9z^2 + 126z + 441 + 16z^2

Combining like terms, we get:

160x^2 + 288x + 144 = 25z^2 + 126z + 441

To simplify this equation further, we need to arrange it in standard form. Subtracting 25z^2 + 126z + 441 from both sides, we have:

160x^2 + 288x + 144 - 25z^2 - 126z - 441 = 0

Combining like terms, we get:

160x^2 + 288x - 25z^2 - 126z - 297 = 0

This is a quadratic equation with respect to x and z. To solve for x and z, we either need additional information or to simplify further.

To solve this problem, let's first define our variables.

Let x represent the length of the shorter leg of the right triangle.

According to the problem, the longer leg is 3 ft longer than three times the shorter leg. So, the longer leg can be represented as 3x + 3.

Also, the hypotenuse is 3 ft shorter than four times the shorter leg. Therefore, the hypotenuse can be expressed as 4x - 3.

Now we can use the Pythagorean theorem to relate the three sides of a right triangle:

(shorter leg)^2 + (longer leg)^2 = (hypotenuse)^2

Substituting the expressions for the sides:

x^2 + (3x + 3)^2 = (4x - 3)^2

Expanding and simplifying the equation:

x^2 + 9x^2 + 18x + 9 = 16x^2 - 24x + 9

Combine like terms:

10x^2 - 42x = 0

Factor out 2x:

2x(5x - 21) = 0

Now we have two possible solutions:

1) 2x = 0 → x = 0
However, a triangle with a side length of 0 does not exist, so we discard this solution.

2) 5x - 21 = 0 → 5x = 21 → x = 21/5

Therefore, the length of the shorter leg is x = 21/5 feet.

Now we can find the lengths of the other two sides:

- The longer leg = 3x + 3 = 3(21/5) + 3 = 63/5 + 15/5 = 78/5 feet.

- The hypotenuse = 4x - 3 = 4(21/5) - 3 = 84/5 - 15/5 = 69/5 feet.

So, the lengths of the three sides of the right triangle are:

Shorter leg = 21/5 feet
Longer leg = 78/5 feet
Hypotenuse = 69/5 feet.

shoter leg = x

longer leg = 3x+3
hypot= 4x-3

to solve for hypotenuse you use
a^2 + b^2 = c^2, so

(x)^2 + (3x+3)^2 = (4x-3)^2

(x)(x) + (3x+3)(3x+3) = (4x-3)(4x-3)
then multiply

(x^2) + (9x^2+18x+9) = (16x^2-24x+9)

then combine like terms

10x^2 + 18x + 9 = 16x^2 - 24x + 9

then i subtracted 10x^2 from both sides
and got:

18x + 9 = 6x^2 - 24x +9

next, i subtracted 18x from both sides:

9 = 6x^2 - 42x + 9

and then subtracted 9 from both sides

0 = 6x^2 - 42x

then pull out a 6x on the right side of the equation

0 = 6x(x-7)

so
6x=0
and
x-7=0

so x= 0 and 7

so the shorter leg is :
0 units or 7 units

the longer leg is :
3 units or 24 units

and the hypotenuse is :
-3 units or 25

and since you cant have a hypotenuse with a negative value
i think the three lengths of your triangle are

7, 24, and 25

Thanks for the help. :)