# CAT Practice : Trigonometry

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When you look at your reflection through a mirror, the image is at a distance equal to the distance between mirror and you. Now, think about what this has to with trigonometry.

## Height and Distances

Q.3: Consider a regular hexagon ABCDEF. There are towers placed at B and D. The angle of elevation from A to the tower at B is 30 degrees, and to the top of the tower at D is 45 degrees. What is the ratio of the heights of towers at B and D?
1. 1 : $\sqrt 3$
2. 1 : 2$\sqrt 3$
3. 1 : 2
4. 3 : 4$\sqrt 3$

Choice B. 1 : 2$\sqrt 3$

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## Detailed Solution

Let the hexagon ABCDEF be of side ‘a’. Line AD = 2a. Let towers at B and D be B’B and D’D respectively.
From the given data we know that ∠B´AB = 30° and ∠D´AB = 45°. Keep in mind that the Towers B’B and D´D are not in the same plane as the hexagon.

In Triangle B’AB,
Tan∠B´AB = ${{{\rm{B'B}}} \over {{\rm{AB}}}}{\rm{ = }}{{\rm{1}} \over {\rm{3}}}$
=> B’B = ${a \over {\sqrt 3 }}$

In Triangle D´AD, tan ∠D´AD = ${{{\rm{D'D}}} \over {{\rm{AD}}}}$ = 1
=> D’D = 2a

Ratio of heights = ${1 \over {\sqrt 3 }}$ : 2 or 1 : 2$\sqrt 3$.

Correct Answer: 1 : 2$\sqrt 3$

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