# Ex.6.5 Q5 The Triangle and Its Properties - NCERT Maths Class 7

## Question

A tree is broken at a height of \(\rm{}5\, m\) from the ground and its top touches the ground at \(\rm{}12\, m\) from the base of the tree. Find the original height of the tree.

## Text Solution

**What is known?**

Height of the point where the tree broke from the ground is \(5 \rm \,m\) and distance between the base of the tree and the top of the tree when it touches the ground is \(\rm{}12\, m \)

**What is unknown?**

Original height of the tree..

**Reasoning:**

This question is also based on the concept of right-angled triangle and Pythagoras theorem. Suppose \(\rm{}P’Q \)is the height of the tree, as it is mentioned in the question that the tree is broken at a height of \(\rm{}5\,m\) from the ground. suppose tree is broken from \(R\) So, consider \(RQ\) as perpendicular and \(PR \) as broken part of the treeand, as hypotenuse. Remember, length of \(PR\) i.e. broken part of the tree will remain same as \(P’R.\) Now, a right – angled triangle \(PQR\) is formed, apply Pythagoras theorem and find the length of broken part i.e. \(PR\). As we must find out the original height of the tree for this, add the length of the broken part and the length where it broke i.e. \(QR.\)

For better visual understanding draw a right-angled triangle and visualise all the parts.

**Steps**

Let \(P’Q \) represents the original height of the tree before it broken at \(R\) and \(RP\) represents the broken part of the tree.

Triangle \(PQR\) is right angled at \(Q\). So, in this triangle, according to Pythagoras theorem,

\(\begin{align}&{{\left( \text{Hypotenuse} \right)}^2} \\&= \rm{{ }}{{\left(\text {Perpendicular} \right)}^2} + {\rm{ }}{{\left( \text{Base} \right)}^2} \end{align} \)

\( \begin{align} {{\left( {PR} \right)}^2} &={{\left( {RQ} \right)}^2} +{{\left( {PQ} \right)}^2}\\{{\left( {PR} \right)}^2} &= \rm{{ }}{{\left( 5 \right)}^2} + {\rm{ }}{{\left( {12} \right)}^2}\\{{\left( {PR} \right)}^2} &= \rm{{ }}25 + 144\\{{\left( {PR} \right)}^2} &= \rm{{ }}169\\PR{\rm{ }} &= \rm{{ }}13{\rm{ }}m\end{align}\)

Thus, the original height of the tree

\[\begin{align}&= PR + RQ \\&= \rm{}13\, m + 5\, m\\&= \rm{}18\,\rm{m} \end{align}\]

Thus, the original height of the tree is \(\rm{}18\, m.\)

**Useful Tip:**

Whenever you encounter problem of this kind, it is best to think of the concept of a right-angled triangle.