Question

A kite flying at a height of 80m from the ground is tied tightly to a nail on the building of height 10m by its thread. If the angle subtended by the thread with the building is \[{30^ \circ }\]. Find the length of the thread.

Answer 1

Hint: We are going to sketch the given values in the form of a diagram with the help of the formulas of height and distance we can find the length of the thread of the kite.

Complete step-by-step answer:

As per the diagram, the length of the building \[AB = 10\]m
That is A is at a height of 10 m from the ground where the thread of kite is tied to a nail.
Also it is given that the kite is flying at a height of \[80\]m from the ground.
So, \[DE = 80\]m
Also the angle subtended by the thread with the building is \[{30^ \circ }\].
So, \[\angle EAC = {30^ \circ }\]
We have to find the length of the thread of the kite
From the diagram the length of the kite is that the length of \[AE.\]
Since,
\[AB = CD = 10\]
Also from the given figure it is clear that, \[CE = DE - CD\]
Substitute the known values in the above equation we get,
\[CE = 80 - 10 = 70\]cm
In the \[\Delta ACE\] using the relation between angle and side we get,
\[\dfrac{{CE}}{{AE}} = \sin {30^ \circ }\]
By rearranging the above equation, we have, \[\dfrac{{CE}}{{\sin {{30}^ \circ }}} = AE\]
Substitute the value we get,
\[\dfrac{{70}}{{\sin {{30}^ \circ }}} = AE\]
On substituting the value of sine in the above equation we get,
\[AE = 140\]
Hence, we have found the length of the thread is 140 m.

$\therefore$ The length of the thread of the kite is \[140\]m.

Note:
For a right-angle triangle, if any one of the angles is \[\theta \], the relation between the angle and the side is \[\sin \theta = \dfrac{{Perpendicular}}{{Hypotenuse}}\]. Don’t confuse between the perpendicular and hypotenuse. Hypotenuse has a larger length than the perpendicular.