A tree is broken at a height of 6 m from the ground and its top touches the ground at a distance of 8m from the tree.Find the original height of the tree.

Options

A. 6 m
B. 16 m
C. 10 m
D. 14 m

Answer

To solve the problem of finding the original height of the tree that broke at a height of 6 meters and whose top touched the ground at a distance of 8 meters from the base of the tree, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**:
- A tree breaks at a height of 6 meters from the ground.
- The top of the tree touches the ground at a distance of 8 meters from the base of the tree.

2. **Visualize the Situation**:
- Let's denote:
- Point A as the point where the tree breaks (6 meters above the ground).
- Point B as the top of the tree (original height).
- Point C as the point where the top of the tree touches the ground.
- The distance from the base of the tree (point O) to where the top touches the ground (point C) is 8 meters.

3. **Use the Right Triangle**:
- The broken part of the tree (from point A to point B) forms a right triangle with:
- The height from point A to the ground (6 meters).
- The horizontal distance from the base of the tree to where the top touches the ground (8 meters).
- The length of the tree from point A to point B is the hypotenuse of this triangle.

4. **Calculate the Length of the Broken Part**:
- We can use the Pythagorean theorem to find the length of the broken part (AB):
\[ AB = \sqrt{(height)^2 + (distance)^2} = \sqrt{(6)^2 + (8)^2} \]
\[ AB = \sqrt{36 + 64} = \sqrt{100} = 10 \text{ meters} \]

5. **Find the Original Height of the Tree**:
- The original height of the tree (OB) is the sum of the height from the ground to the break (OA) and the length of the broken part (AB):
\[ OB = OA + AB = 6 \text{ meters} + 10 \text{ meters} = 16 \text{ meters} \]

6. **Conclusion**:
- The original height of the tree is **16 meters**.