This problem involves finding the hypotenuse $x$ of a right triangle using the Pythagorean theorem. The sides given are:

• One leg: $5$
• Another leg: $6 + 1 = 7$

Solution:

For a right triangle with legs $a$ and $b$, and hypotenuse $c$, the Pythagorean theorem states: $a^2 + b^2 = c^2$

In this case: $5^2 + 7^2 = x^2$

Calculating each term: $5^2 = 25$ $7^2 = 49$

Adding these: $25 + 49 = 74$

So, $x^2 = 74$

To solve for $x$, take the square root of both sides: $x = \sqrt{74}$

Since $\sqrt{74}$ is already in its simplest radical form, the answer is: $x = \sqrt{74}$

Would you like more details or have any questions?

Here are five related questions that build on this concept:

1. How would you solve for a missing leg if the hypotenuse and one leg are known?
2. What are some methods to approximate $\sqrt{74}$ to a decimal value?
3. How can you simplify a square root if the number under the root has factors that are perfect squares?
4. What are some real-life applications of the Pythagorean theorem?
5. How does the Pythagorean theorem extend to three dimensions?

Tip: Always double-check the lengths of all sides in a triangle to ensure they meet the conditions needed for applying the Pythagorean theorem.