This problem involves a ladder leaning against a wall, which forms a right triangle. To find the length of the ladder, we can apply the Pythagorean Theorem:

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

Where:

• $a = 10 \, \text{ft}$ (the distance from the house at the bottom),
• $b = 19 \, \text{ft}$ (the height the ladder reaches up the wall),
• $c$ is the length of the ladder (the hypotenuse, which we need to find).

Let's solve for $c$:

$c = \sqrt{a^2 + b^2} = \sqrt{10^2 + 19^2} = \sqrt{100 + 361} = \sqrt{461}$

Now, I'll calculate the exact value and round it to the nearest tenth.The length of the ladder is approximately 21.5 feet when rounded to the nearest tenth.

Would you like any further details or have any questions?

Here are some related questions you might find useful:

1. What is the Pythagorean theorem and how is it used in different scenarios?
2. How do you solve for unknown sides in right triangles with trigonometry?
3. How do different types of triangles affect the application of the Pythagorean theorem?
4. How can the ladder problem be solved if we know angles instead of side lengths?
5. How is the Pythagorean theorem applied in real-world construction problems?

Tip: The Pythagorean theorem only works for right triangles, where one angle is exactly 90 degrees.