Question

A boy standing at a point O finds his kite flying at a point P with distance OP=25m. It is at a height 5m from the ground. When thread is extended by 10m from P it reaches the point Q. What will be the height QN of the kite from the ground? (Use the trigonometric ratios)

Answer 1

Hint: We solve this problem by considering the triangle OPM and then taking the sine of the angle POM. Then we consider the triangle OQN and apply sine to the angle QON and then we equate the two values as the angles POM and QON are equal. Then we solve the obtained equation to find the value of h, that is the height of the kite from the ground.

Complete step by step answer:
Let us consider the formula for sine of an angle in a right-angled triangle.
$\sin \theta =\dfrac{\text{Opposite Side}}{\text{Hypotenuse}}$
Now, let us consider the triangle OPM. We can see that it is a right-angled triangle. So, applying the sine formula for angle POM we get,
$\sin \left( \angle POM \right)=\dfrac{PM}{OP}$
Now, let us substitute the values of sides OP and PM in the above formula. Then we get,
$\begin{align}
  & \Rightarrow \sin \left( \angle POM \right)=\dfrac{5}{25} \\
 & \Rightarrow \sin \left( \angle POM \right)=\dfrac{1}{5}..............\left( 1 \right) \\
\end{align}$
Now, let us consider the triangle OQN.
We can write the angle QON as
$\sin \left( \angle QON \right)=\dfrac{QN}{OQ}$
As we can write OQ=OP+PQ, we get
$OQ=25+10=35$
Now, let us substitute the values of sides OP and PM in the above formula. Then we get,
$\Rightarrow \sin \left( \angle QON \right)=\dfrac{h}{35}............\left( 2 \right)$
We can see that the angels QON and POM are equal. So, by equating the values in equation (1) and equation (2) we get,
$\begin{align}
  & \Rightarrow \sin \left( \angle QON \right)=\sin \left( \angle POM \right) \\
 & \Rightarrow \dfrac{h}{35}=\dfrac{1}{5} \\
 & \Rightarrow h=\dfrac{35}{5} \\
 & \Rightarrow h=7 \\
\end{align}$
So, we get that the length of QN as QN=7m.

Hence, the correct answer is 7m.

Note: While solving this problem one might confuse and make a mistake by taking the length of the side OQ as 10m. But it is wrong as we are given that the thread is extended 10m from the point P not from O. So, we get the length of OQ as 25+10, that is 35m.