solve the measurement of the 3 sides of the triangle;

the shorter leg is x, the longer leg is 2x+2, the hypotenuse id 2x+3

We can solve for the sides of the triangle using the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (\(c\)) is equal to the sum of the squares of the other two sides (\(a\) and \(b\)):

\[
c^2 = a^2 + b^2
\]

Given:
- Shorter leg: \(x\)
- Longer leg: \(2x + 2\)
- Hypotenuse: \(2x + 3\)

Using the Pythagorean theorem:

\[
(2x + 3)^2 = x^2 + (2x + 2)^2
\]

Now, let's solve for \(x\).

1. Expand both sides:
\[
(2x + 3)^2 = (2x + 3)(2x + 3) = 4x^2 + 12x + 9
\]
\[
x^2 + (2x + 2)^2 = x^2 + (4x^2 + 8x + 4)
\]

2. Combine and simplify:
\[
4x^2 + 12x + 9 = x^2 + 4x^2 + 8x + 4
\]
\[
4x^2 + 12x + 9 = 5x^2 + 8x + 4
\]

3. Rearrange the equation:
\[
0 = x^2 - 4x - 5
\]

4. Factor the quadratic equation:
\[
(x - 5)(x + 1) = 0
\]

So, \(x = 5\) or \(x = -1\).

Since \(x\) represents the length of a side of a triangle, it must be positive, so \(x = 5\).

Finally, the sides of the triangle are:
- Shorter leg: \(x = 5\)
- Longer leg: \(2x + 2 = 2(5) + 2 = 12\)
- Hypotenuse: \(2x + 3 = 2(5) + 3 = 13\)

### The sides of the triangle are:
- Shorter leg: 5
- Longer leg: 12
- Hypotenuse: 13

Would you like further details, or do you have any questions?

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Here are five related questions:
1. How would the solution change if the longer leg were \(3x + 1\)?
2. What is the significance of the Pythagorean triple in this problem?
3. How can you verify the solution by checking the perimeter of the triangle?
4. What if the hypotenuse was \(2x + 4\)? How would that affect the solution?
5. How can the Pythagorean theorem be applied to non-right triangles?

**Tip:** When solving quadratic equations, remember to consider only the positive root when the variable represents a physical quantity like length.

Learn how to find the sides of a triangle using the Pythagorean theorem with given lengths x, 2x+2, and 2x+3. This problem involves solving a quadratic equation and verifying triangle properties. Suitable for high school students.

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