This is a classic right triangle problem, where we can apply the Pythagorean Theorem.

The Pythagorean Theorem states that for a right triangle:

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

where:

• $a$ is the height the ladder reaches on the house,
• $b$ is the horizontal distance from the base of the ladder to the house, and
• $c$ is the length of the ladder (the hypotenuse).

Given:

• $c = 24$ feet (length of the ladder),
• $b = 13$ feet (distance from the base of the ladder to the house),
• $a$ is what we need to find (the height the ladder reaches on the house).

Now, apply the Pythagorean theorem:

$a^2 + 13^2 = 24^2$

First, calculate $13^2$ and $24^2$:

$a^2 + 169 = 576$

Now, subtract 169 from both sides:

$a^2 = 576 - 169$

$a^2 = 407$

Now, take the square root of both sides to find $a$:

$a = \sqrt{407} \approx 20.2 \text{ feet}$

So, the ladder reaches approximately 20.2 feet up the side of the house.

Would you like more details or have any questions? Here are some related questions for further exploration:

1. How would the height change if the ladder was placed 12 feet away from the house?
2. How does the Pythagorean theorem apply in non-right triangle scenarios?
3. What other real-life scenarios can we use the Pythagorean theorem in?
4. If the ladder was longer, say 30 feet, how would that affect the height reached?
5. How can we calculate the angle the ladder makes with the ground?

Tip: When solving for missing sides in a right triangle using the Pythagorean theorem, always make sure you have identified the hypotenuse correctly (the longest side).