This problem requires solving for the lengths of a right triangle's sides using the Pythagorean theorem, which states:

$a^2 + b^2 = c^2$

Here:

• $a = x$ (the shorter leg),
• $b = x + 7$ (the longer leg),
• $c = 2x + 3$ (the hypotenuse).

Step 1: Substitute into the Pythagorean theorem:

$x^2 + (x + 7)^2 = (2x + 3)^2$

Step 2: Expand the terms:

1. Expand $(x + 7)^2$: $(x + 7)^2 = x^2 + 14x + 49$

2. Expand $(2x + 3)^2$: $(2x + 3)^2 = 4x^2 + 12x + 9$

Now, substitute these into the equation: $x^2 + (x^2 + 14x + 49) = 4x^2 + 12x + 9$

Step 3: Simplify:

Combine like terms on the left-hand side: $2x^2 + 14x + 49 = 4x^2 + 12x + 9$

Rearrange to set the equation to 0: $0 = 4x^2 + 12x + 9 - 2x^2 - 14x - 49$

$0 = 2x^2 - 2x - 40$

Factor out 2: $x^2 - x - 20 = 0$

Step 4: Factorize:

Solve $x^2 - x - 20 = 0$: $(x - 5)(x + 4) = 0$

Step 5: Solve for $x$:

$x = 5 \quad \text{or} \quad x = -4$

Since lengths cannot be negative: $x = 5$

Step 6: Calculate the side lengths:

1. $x = 5$,
2. Longer leg: $x + 7 = 5 + 7 = 12$,
3. Hypotenuse: $2x + 3 = 2(5) + 3 = 13$.

Final Answer:

The side lengths are: $\boxed{5, 12, 13}$

Would you like further details or explanations?

Related Questions:

1. How is the Pythagorean theorem applied in real-life scenarios?
2. What are the properties of a 5-12-13 triangle?
3. Can you explain how quadratic equations are solved using the quadratic formula?
4. How do you determine if a triangle is a right triangle given its sides?
5. What other methods can be used to solve quadratic equations?

Tip:

For problems involving quadratic equations, always check for factoring possibilities before using the quadratic formula—it’s often faster!