A tree is broken by wind, its upper part touches the ground at a point 10 metres from the foot of the tree and makes an angle of $45^{\circ}$ with the ground. The entire length of the tree is:

Question

Accepted Answer

To determine the entire length of the tree, we need to analyze the situation using trigonometry. The problem states that the upper part of the tree touches the ground at a point 10 meters from the foot of the tree and makes an angle of $45^\circ$ with the ground.

### Explanation:

1. **Understanding the Problem:**
   - The tree is broken, and the upper part forms a right triangle with the ground.
   - The horizontal distance from the foot of the tree to the point where the top touches the ground is 10 meters.
   - The angle between the ground and the broken part of the tree is $45^\circ$.

2. **Using Trigonometry:**
   - In a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side.
   - Here, the tangent of $45^\circ$ is 1 (since $\tan 45^\circ = 1$).
   - Therefore, the length of the broken part of the tree (opposite side) is equal to the horizontal distance (adjacent side), which is 10 meters.

3. **Calculating the Entire Length of the Tree:**
   - The entire length of the tree is the sum of the unbroken part (vertical part) and the broken part.
   - Since the angle is $45^\circ$, the vertical part is also 10 meters (as $\tan 45^\circ = 1$).
   - Therefore, the length of the tree is the hypotenuse of the right triangle formed, which can be calculated using the Pythagorean theorem or recognizing the properties of a $45^\circ$ triangle.

4. **Using the Pythagorean Theorem:**
   - The hypotenuse $h$ of a right triangle with equal legs (both 10 meters) is given by:
     $$
     h = \sqrt{10^2 + 10^2} = \sqrt{200} = 10\sqrt{2}
     $$
   - The entire length of the tree is the sum of the vertical part and the hypotenuse:
     $$
     \text{Total length} = 10 + 10\sqrt{2} = 10(1 + \sqrt{2})
     $$

5. **Conclusion:**
   - Therefore, the entire length of the tree is $10(1 + \sqrt{2})$ meters, which corresponds to Option 4.

### Why Other Options are Incorrect:

- **Option 1: $10(1+\frac{\sqrt{3}}{2})$ metres**
  - This option incorrectly uses $\frac{\sqrt{3}}{2}$, which is not relevant to a $45^\circ$ angle. This value is more related to angles like $30^\circ$ or $60^\circ$.

- **Option 2: 20 metres**
  - This option assumes the tree's length is simply the sum of the horizontal and vertical parts without considering the hypotenuse formed by the broken part.

- **Option 3: 30 metres**
  - This option overestimates the length by assuming a different configuration or miscalculating the components of the triangle.

By following the trigonometric principles and correctly applying them to the given scenario, Option 4 is confirmed as the correct answer.