Discussion
A person standing exactly midway between two towers observes their tops at angles of elevation of \( 22.5^{\circ} \) and \( 67.5^{\circ} \). What is the ratio of the height of the taller tower to that of the shorter one? (Given \( \tan 22.5^{\circ} = \sqrt{2} - 1 \))
Answer: D
Let the observer be \( d \) from each tower, and let the heights be \( h_{1} \) (at \( 67.5^{\circ} \)) and \( h_{2} \) (at \( 22.5^{\circ} \)).
\[ \frac{h_{1}}{h_{2}} = \frac{d\tan 67.5^{\circ}}{d\tan 22.5^{\circ}} = \frac{\tan 67.5^{\circ}}{\tan 22.5^{\circ}} \]
The two angles are complementary, so
\[ \tan 67.5^{\circ} = \tan(90^{\circ} - 22.5^{\circ}) = \cot 22.5^{\circ} = \frac{1}{\sqrt{2}-1} = \sqrt{2}+1 \]
\[ \frac{h_{1}}{h_{2}} = \frac{\sqrt{2}+1}{\sqrt{2}-1} = (\sqrt{2}+1)^{2} = 2 + 2\sqrt{2} + 1 = 3 + 2\sqrt{2} \]
So the ratio is \( (3 + 2\sqrt{2}) : 1 \), about \( 5.83 : 1 \). The distance \( d \) cancels, which is why "exactly midway" is all the question needs to say.
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