To solve this problem, let's break it down step by step.

Problem Summary:

• A tree has broken due to a storm.
• The top of the broken tree touches the ground, forming a right triangle.
• The angle between the tree top and the ground is $45^\circ$.
• The distance from the foot of the tree to the point where the top touches the ground is 10 meters.

We need to find the original height of the tree.

Solution:

In the right triangle formed:

• Let $L$ be the length of the broken part of the tree (the hypotenuse of the triangle).
• The distance between the foot of the tree and the point where the top touches the ground is $10$ meters (this is the base of the triangle).
• Since the angle is $45^\circ$, the base and the height of the triangle are equal. Therefore, the height $h$ of the tree from the ground to the break point is also $10$ meters.

Using the Pythagorean theorem for a 45-degree right triangle: $L = \frac{\text{Base}}{\cos 45^\circ} = \frac{10}{\cos 45^\circ} = \frac{10}{\frac{1}{\sqrt{2}}} = 10 \times \sqrt{2} = 10 \times 1.414 \approx 14.14 \text{ meters}$

So, the length of the broken part of the tree $L$ is approximately 14.14 meters.

Original Height of the Tree:

The original height of the tree before it broke is the sum of the height of the part still standing and the length of the broken part: $\text{Total Height} = h + L = 10 + 14.14 \approx 24.14 \text{ meters}$

Final Answer:

The original height of the tree was approximately 24.14 meters.

Would you like more details or have any questions?

5 Related Questions:

1. What is the significance of the 45-degree angle in this problem?
2. How would the problem change if the angle were 60 degrees instead of 45 degrees?
3. How can the Pythagorean theorem be applied in other