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 metres
B. 20 metres
C. `10(1+sqrt2)` metres
D. `10(1+sqrt3/2)` metres

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the problem
We have a tree that is broken by the wind. The upper part of the tree touches the ground at a point 10 meters away from the foot of the tree and makes an angle of 45 degrees with the ground.

### Step 2: Draw a diagram
Draw a right triangle where:
- Point A is the top of the tree.
- Point B is the foot of the tree.
- Point C is the point where the upper part of the tree touches the ground.
- The distance BC (the horizontal distance from the foot of the tree to where the tree touches the ground) is 10 meters.
- The angle CAB (the angle between the ground and the broken part of the tree) is 45 degrees.

### Step 3: Use trigonometric ratios
In triangle ABC, we can use the tangent function because we know the angle and one side. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side.

\[ \tan(45^\circ) = \frac{BC}{AB} \]

Since \(\tan(45^\circ) = 1\), we can set up the equation:

\[ 1 = \frac{BC}{AB} \]

### Step 4: Solve for BC
Since BC is given as 10 meters, we can substitute:

\[ 1 = \frac{10}{AB} \implies AB = 10 \text{ meters} \]

### Step 5: Calculate AC using Pythagorean theorem
Now, we need to find the length of the broken part of the tree, AC. In triangle ABC, we can use the Pythagorean theorem:

\[ AC^2 = AB^2 + BC^2 \]

Substituting the values we have:

\[ AC^2 = (10)^2 + (10)^2 \]
\[ AC^2 = 100 + 100 = 200 \]
\[ AC = \sqrt{200} = 10\sqrt{2} \text{ meters} \]

### Step 6: Find the total length of the tree
The total length of the tree (AB + AC) is:

\[ AB + AC = 10 + 10\sqrt{2} \]

### Final Answer
The entire length of the tree is:

\[ 10(1 + \sqrt{2}) \text{ meters} \]