A tree is broken at a height of 5 m from the groundand its top touches the ground at a distance of 12 m from the base of the tree. Find the original height of the tree

Options

A. `17m`
B. `18m`
C. `16m`
D. `14m`

Explanation

To find the original height of the tree, we can follow these steps:

### Step 1: Understand the Problem
The tree is broken at a height of 5 meters from the ground, and the top of the tree touches the ground at a distance of 12 meters from the base of the tree. We need to find the original height of the tree.

### Step 2: Draw a Diagram
Draw a right triangle where:
- Point A is the top of the tree.
- Point B is the point where the tree is broken (5 meters above the ground).
- Point C is the base of the tree.
- The distance from B to C is 12 meters (the distance from the base of the tree to where the top touches the ground).
- The distance from B to A is the height of the tree above the break, which is 5 meters.

### Step 3: Identify the Right Triangle
In the right triangle ABC:
- AB (the height from the break to the top of the tree) = 5 meters (perpendicular).
- BC (the distance from the base of the tree to where the top touches the ground) = 12 meters (base).
- AC (the length of the broken part of the tree that touches the ground) = x (hypotenuse).

### Step 4: Apply the Pythagorean Theorem
According to the Pythagorean theorem:
\[ AC^2 = AB^2 + BC^2 \]
Substituting the known values:
\[ x^2 = 5^2 + 12^2 \]

### Step 5: Calculate the Squares
Calculate \( 5^2 \) and \( 12^2 \):
- \( 5^2 = 25 \)
- \( 12^2 = 144 \)

### Step 6: Add the Squares
Now, add these values:
\[ x^2 = 25 + 144 \]
\[ x^2 = 169 \]

### Step 7: Take the Square Root
Now take the square root of both sides to find x:
\[ x = \sqrt{169} \]
\[ x = 13 \text{ meters} \]

### Step 8: Find the Original Height of the Tree
The original height of the tree (let's denote it as H) is the sum of the height above the break (AB) and the length of the broken part (AD):
\[ H = AB + AD \]
Where:
- \( AB = 5 \text{ meters} \)
- \( AD = 13 \text{ meters} \)

So,
\[ H = 5 + 13 = 18 \text{ meters} \]

### Final Answer
The original height of the tree is **18 meters**.

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