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We are given a right triangle where:

1. The shorter leg is 5 feet shorter than the longer leg.
2. The hypotenuse is 5 feet longer than the longer leg.

Let's define the variables:
- Let the length of the longer leg be \( x \).
- Then the shorter leg is \( x - 5 \).
- The hypotenuse is \( x + 5 \).

Using the Pythagorean Theorem, which states:
\[
\text{(Shorter leg)}^2 + \text{(Longer leg)}^2 = \text{(Hypotenuse)}^2
\]
Substitute the values:
\[
(x - 5)^2 + x^2 = (x + 5)^2
\]
Now expand and solve the equation:
\[
(x^2 - 10x + 25) + x^2 = (x^2 + 10x + 25)
\]
Simplify the equation:
\[
2x^2 - 10x + 25 = x^2 + 10x + 25
\]
Subtract \( x^2 + 10x + 25 \) from both sides:
\[
x^2 - 20x = 0
\]
Factor the equation:
\[
x(x - 20) = 0
\]
So, \( x = 20 \) (since \( x = 0 \) isn't physically meaningful in this context).

Thus:
- The longer leg is \( x = 20 \) feet.
- The shorter leg is \( x - 5 = 20 - 5 = 15 \) feet.
- The hypotenuse is \( x + 5 = 20 + 5 = 25 \) feet.

### Final Answer:
- Length of the shorter leg: **15 ft**
- Length of the longer leg: **20 ft**
- Length of the hypotenuse: **25 ft**

Would you like a detailed breakdown of any part of this solution?

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Here are 5 related questions to explore further:
1. What would happen if the hypotenuse were 10 feet longer than the longer leg instead?
2. How can we apply the Pythagorean Theorem in non-right triangles?
3. What are the different properties of a 3-4-5 triangle?
4. Can a right triangle have legs with non-integer lengths and still satisfy the Pythagorean Theorem?
5. How would this problem change if the shorter leg were given instead of the longer leg?

**Tip**: Always remember that in any right triangle, the hypotenuse is always the longest side.

The shorter leg of a right triangle is 5 ft shorter than the longer leg. The hypotenuse is 5 ft longer than the longer leg. Find the side lengths of the triangle.

Solve a right triangle problem where the shorter leg is 5 feet shorter than the longer leg, and the hypotenuse is 5 feet longer. Learn how to apply the Pythagorean Theorem and quadratic equations in this geometry problem, suitable for Grades 8-10.

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