A tree of height 'h' metres is broken by a storm in such a way that its top touches the ground at a distance of 'x' metres from its root. Find the height at which the tree is broken. (Here `h gt x`)

Options

A. A)`(h^2 + x^2)/(2h)` metre
B. B)`(h^2 - x^2)/(2h)` metre
C. C)`(h^2 + x^2)/(4h)` metre
D. D)`(h^2 - x^2)/(4h)` metre

Answer

To solve the problem of finding the height at which the tree is broken, we can follow these steps:

### Step 1: Understand the Problem
We have a tree of height \( h \) meters that is broken at a certain height \( y \). The top of the tree now touches the ground at a distance \( x \) meters from the base of the tree.

### Step 2: Set Up the Triangle
When the tree breaks, it forms a right triangle:
- The height of the broken part of the tree is \( y \) (the height at which the tree is broken).
- The distance from the base of the tree to the point where the top touches the ground is \( x \).
- The length of the broken tree from the break point to the top is \( h - y \).

### Step 3: Apply the Pythagorean Theorem
In the right triangle formed, we can apply the Pythagorean theorem:
\[ AB^2 + BC^2 = AC^2 \]
Where:
- \( AB = y \) (the height at which the tree is broken),
- \( BC = x \) (the distance from the base to the point where the top touches the ground),
- \( AC = h - y \) (the remaining height of the tree).

Substituting these values into the equation gives:
\[ y^2 + x^2 = (h - y)^2 \]

### Step 4: Expand the Equation
Now, we expand the right side of the equation:
\[ y^2 + x^2 = h^2 - 2hy + y^2 \]

### Step 5: Simplify the Equation
Next, we can simplify the equation by canceling \( y^2 \) from both sides:
\[ x^2 = h^2 - 2hy \]

### Step 6: Rearrange to Solve for \( y \)
Rearranging the equation gives:
\[ 2hy = h^2 - x^2 \]
Now, divide both sides by \( 2h \):
\[ y = \frac{h^2 - x^2}{2h} \]

### Final Answer
Thus, the height at which the tree is broken is:
\[ y = \frac{h^2 - x^2}{2h} \]