To solve this problem, we can use the Pythagorean theorem, which relates the sides of a right triangle:

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

where:

• $a$ is the distance from the base of the tree to the bottom of the ladder (6 feet),
• $b$ is the height the ladder reaches on the tree (which we want to find),
• $c$ is the length of the ladder (10 feet).

We know $a = 6$ feet and $c = 10$ feet. To find $b$, we substitute the known values into the Pythagorean theorem:

$6^2 + b^2 = 10^2$ $36 + b^2 = 100$ $b^2 = 100 - 36$ $b^2 = 64$ $b = \sqrt{64}$ $b = 8 \text{ feet}$

So, the ladder reaches 8 feet up the tree.

Answer: 8 feet

Let me know if you'd like further details! Here are a few related questions you might find interesting:

1. How would the height change if the bottom of the ladder was placed 5 feet away from the tree?
2. What is a Pythagorean triple?
3. If the ladder were 12 feet long, how high could it reach with the same 6 feet distance?
4. What happens if the ladder is placed directly against the tree (distance is 0 feet)?
5. How can the Pythagorean theorem be applied to find distances in real-world scenarios like this?

Tip: The Pythagorean theorem only works for right-angled triangles, where one angle is 90°.