The image shows a right triangle, where one leg measures 6 inches, the other leg is split into 4 inches and $x$, and the hypotenuse is 10 inches.

We can apply the Pythagorean theorem, which states: $a^2 + b^2 = c^2$ where $a$ and $b$ are the legs, and $c$ is the hypotenuse.

In this case:

• One leg (the total horizontal distance) is $6 + x$.
• The other leg (vertical) is 4 inches.
• The hypotenuse is 10 inches.

We can plug these into the Pythagorean theorem: $(6 + x)^2 + 4^2 = 10^2$ Now, let's solve for $x$.The solution to the equation gives two possible values for $x$:

$x = -6 + 2\sqrt{21} \quad \text{or} \quad x = -6 - 2\sqrt{21}$

However, since $x$ represents a length, it cannot be negative. Therefore, we only consider the positive value:

$x = -6 + 2\sqrt{21} \approx 2.17$

This is approximately 2.17 inches, but it doesn't match any of the provided choices exactly. There might be an error in rounding or assumptions. Would you like to explore more details on this or clarify further?

Follow-up questions:

1. How would you handle cases where the computed answer doesn't match the given choices exactly?
2. Can the Pythagorean theorem be applied in cases of obtuse triangles?
3. What does the square root operation represent geometrically in this context?
4. How can we refine estimates when dealing with irrational numbers like $\sqrt{21}$?
5. In what other real-life situations can the Pythagorean theorem be applied?

Tip: Always double-check your assumptions in geometry problems, particularly with lengths and distances.