To solve this problem, we can follow these steps based on the given conditions:

Let:

• $x$ be the length of the shorter leg of the right triangle.
• The longer leg is $x + 3$ feet (since it is 3 feet longer than the shorter leg).
• The hypotenuse is $x + 6$ feet (since it is 6 feet longer than the shorter leg).

Since this is a right triangle, we can use the Pythagorean theorem:

$\text{(shorter leg)}^2 + \text{(longer leg)}^2 = \text{(hypotenuse)}^2$

Substituting the values:

$x^2 + (x + 3)^2 = (x + 6)^2$

Now, let's solve this equation to find $x$.

$x^2 + (x^2 + 6x + 9) = (x^2 + 12x + 36)$

Simplifying:

$2x^2 + 6x + 9 = x^2 + 12x + 36$

Move all terms to one side of the equation:

$2x^2 + 6x + 9 - x^2 - 12x - 36 = 0$

Simplify further:

$x^2 - 6x - 27 = 0$

Now, solve this quadratic equation using the quadratic formula:

$x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(-27)}}{2(1)}$

$x = \frac{6 \pm \sqrt{36 + 108}}{2}$

$x = \frac{6 \pm \sqrt{144}}{2}$

$x = \frac{6 \pm 12}{2}$

So, we get two possible solutions:

$x = \frac{6 + 12}{2} = 9 \quad \text{or} \quad x = \frac{6 - 12}{2} = -3$

Since the length cannot be negative, the shorter leg $x = 9$ feet.

Thus:

• The shorter leg is $9$ feet.
• The longer leg is $9 + 3 = 12$ feet.
• The hypotenuse is $9 + 6 = 15$ feet.

Final answer:

• Length of the shorter leg: 9 feet
• Length of the longer leg: 12 feet
• Length of the hypotenuse: 15 feet

Would you like more details or have any questions?

Related Questions:

1. How does the Pythagorean theorem work for non-right triangles?
2. How can the quadratic formula be used for different types of quadratic equations?
3. How would the problem change if the hypotenuse was shorter than the longer leg?
4. Can you explain a different method to solve this triangle problem?
5. How can we check if a triangle is a right triangle using only its sides?

Tip:

For any right triangle problem, always consider using the Pythagorean theorem, and if you have relationships between the sides, set up an equation to solve them efficiently.