Question

The top of a broken tree touches the ground at a distance of $12\mathrm{m}$ from its base. If the tree is broken at a height of $5\mathrm{m}$ from the ground then the actual height of the tree is

• A. $25\mathrm{m}$
• B. $13\mathrm{m}$
• C. $18\mathrm{m}$
• D. $17\mathrm{m}$

Answer 1

The correct option is C

$18\mathrm{m}$

Explanation for the correct option:

Given, the top of a broken tree touches the ground at a distance of $12\mathrm{m}$ from its base.

The tree is broken at a height of $5\mathrm{m}$ from the ground.

We have to find the actual height of the tree.

Let $\mathrm{AB}$ be the height of the tree.

$\mathrm{AB}=\mathrm{hm}$

BC be the height from the ground where the tree is broken

$\mathrm{BC}=5\mathrm{m}$

BD is the distance from the base and top of the tree where it touches the ground

$\mathrm{BD}=12\mathrm{m}$

We know that $\mathrm{AC}=\mathrm{CD}$, length of the broken tree.

$$\begin{gathered} \mathrm{AB}=AC+BC \\ \Rightarrow \mathrm{h}=\mathrm{AC}+5 \\ \Rightarrow \mathrm{AC}=\mathrm{h}-5 \\ \therefore \mathrm{CD}=\mathrm{h}-5 \end{gathered}$$

By using the Pythagorean theorem,

$$\begin{gathered} C{D}^2=A{B}^2+B{D}^2 \\ \Rightarrow {(h-5)}^2=5^2+{12}^2 \\ \Rightarrow {(h-5)}^2=25+144 \\ \therefore {(h-5)}^2=169 \end{gathered}$$

Taking square root,

$$\begin{gathered} \Rightarrow \mathrm{h}-5=13 \\ \Rightarrow \mathrm{h}=13+5 \\ \therefore \mathrm{h}=18\mathrm{m} \end{gathered}$$

Therefore, the height of the tree is $18\mathrm{m}$.

Hence, the correct option is (C).