1. **State the problem:** We have a ladder of length 10 m leaning against a tree. The foot of the ladder is 2.5 m away from the tree. We want to find how high up the tree the cat is. 2. **Formula used:** We use the Pythagorean theorem for right triangles, which states: $$a^2 + b^2 = c^2$$ where $c$ is the hypotenuse (ladder length), and $a$ and $b$ are the other two sides (height up the tree and distance from the tree). 3. **Assign values:** Let $h$ be the height up the tree (unknown), the distance from the tree is 2.5 m, and the ladder length is 10 m. 4. **Apply the formula:** $$h^2 + 2.5^2 = 10^2$$ 5. **Calculate:** $$h^2 + 6.25 = 100$$ 6. **Isolate $h^2$:** $$h^2 = 100 - 6.25$$ $$h^2 = 93.75$$ 7. **Find $h$ by taking the square root:** $$h = \sqrt{93.75}$$ 8. **Simplify the square root:** $$h \approx 9.68$$ **Final answer:** The cat is approximately 9.68 meters high up the tree.