Question

A $40$ m high tower is $9$ m away from the point on the ground where there is a hook. A small pole is also standing in the exact middle of the tower and the hook. Now, a rope is tied to the top of the tower and to the hook on the ground, which is also touching the top of the pole.

The length of the middle pole and the rope tied to the tower and the hook are, respectively -

• A. $20$ m and $41$ m
• B. $25$ m and $45$ m
• C. $30$ m and $44$ m
• D. None

Answer 1

The correct option is A $20$ m and $41$ m

The end points of the tower, the pole and the position of the hook are marked as shown in figure.


Now, in triangles $△ABE$ and $△CDE$,
$\angle E=\angle E$ (common angle)
$\angle B=\angle$ $D$ (both ${90}^o$)

So, $△ABE\sim △CDE$
(by AA property)

Given: $\frac{ED}{BE}=\frac{1}{2}$
As sides of similar triangles are in proportion,
$\frac{ED}{BE}=\frac{1}{2}=\frac{CD}{AB}$

Given: $AB=40m$
So, $CD=\frac{1}{2}\times 40m$
$\Rightarrow CD=20m$

In $△$ $ABE,$
applying Pythagoras theorem,
$A{E}^2=A{B}^2+B{E}^2$
$A{E}^2={40}^2+9^2$
$A{E}^2=1600+81$
$A{E}^2=1681$
$AE=\sqrt{1681}$
$AE=41m$
Hence, height of middle pole is $20m$, and the length of rope tied to the tower is $41m.$

So, option $\left(a\right)$ correct.