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^@` with the ground . The entire length of the tree is

Options

A. 15 m
B. 20 m
C. `10(1+sqrt(2))`m
D. `10(1+sqrt(3)/2)`m

Answer

To solve the problem step by step, we will analyze the situation involving the broken tree and apply trigonometric concepts.

### Step 1: Understand the Problem
We have a tree that has been broken by the wind. The upper part of the tree touches the ground at a point that is 10 meters away from the base of the tree. The angle formed between the ground and the broken part of the tree is 45 degrees.

### Step 2: Draw a Diagram
Let's label the points:
- Let point A be the top of the tree (the broken part).
- Let point B be the point where the tree was originally standing (the foot of the tree).
- Let point C be the point where the top of the tree touches the ground.

The distance BC (the horizontal distance from the foot of the tree to the point where the top touches the ground) is given as 10 meters.

### Step 3: Identify the Triangle
In triangle ABC:
- AB is the height of the tree above the break point.
- BC is the distance from the foot of the tree to the point where the tree touches the ground (10 meters).
- AC is the length of the broken part of the tree.

### Step 4: Use Trigonometry
Since we know the angle at C (angle ACB) is 45 degrees, we can use the tangent function:
\[ \tan(45^\circ) = \frac{AB}{BC} \]
Since \(\tan(45^\circ) = 1\), we have:
\[ 1 = \frac{AB}{10} \]
This implies:
\[ AB = 10 \text{ meters} \]

### Step 5: Calculate the Length of AC
Now we will use the Pythagorean theorem to find the length of AC:
\[ AC^2 = AB^2 + BC^2 \]
Substituting the known values:
\[ AC^2 = 10^2 + 10^2 \]
\[ AC^2 = 100 + 100 = 200 \]
\[ AC = \sqrt{200} = 10\sqrt{2} \text{ meters} \]

### Step 6: Find the Total Length of the Tree
The total length of the tree (original height before it broke) is the sum of AB and AC:
\[ \text{Total Length} = AB + AC = 10 + 10\sqrt{2} \]
Thus, the entire length of the tree is:
\[ \text{Total Length} = 10(1 + \sqrt{2}) \text{ meters} \]

### Final Answer
The entire length of the tree is \(10(1 + \sqrt{2})\) meters.

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