바위섬까지의 거리

[출제의도] 삼각비를 이해하여 삼각형에서 선분의 길이를 구한다.

$\displaystyle \mathrm{\overline{CH}} = \mathit{k}$라 하면 직각삼각형 $\mathrm{A} HC$에서 $\displaystyle \tan 60 ^\circ = \frac{{\overline{\mathrm{AH}}}}{{\overline{\mathrm{CH}}}}$ $\displaystyle {\overline{\mathrm{AH}}}$$\displaystyle \mathrm{=} \tan 60 ^\circ \times {\overline{\mathrm{CH}}}$$\displaystyle = \sqrt{3} k$ $\cdots$㉠ 삼각형 $\mathrm{A} BH$는 이등변삼각형이므로 $\displaystyle \overline{\mathrm{A} H} = \overline{\mathrm{B} H}$ ㉠에 의하여 $\displaystyle \mathrm{\overline{BH} = \overline{AH} = \sqrt{3} \mathit{k}}$ $\displaystyle \overline{\mathrm{B} C} = \overline{\mathrm{B}} H + \overline{\mathrm{H} C}$이므로 $\displaystyle \overline{\mathrm{B} C} = 200$에서 $\displaystyle \sqrt{3} k + k = ( \sqrt{3} + 1 ) k = 200$ 그러므로 $k$$\displaystyle = \frac{200}{\sqrt{3} + 1}$$\displaystyle = \frac{200 ( \sqrt{3} - 1 )}{( \sqrt{3} + 1 ) ( \sqrt{3} - 1 )}$$\displaystyle = 100 ( \sqrt{3} - 1 )$ 따라서 구하는 선분 $\mathrm{A} H$의 길이는 $\displaystyle {\overline{\mathrm{AH}}}$$\displaystyle = \sqrt{3} k$$\displaystyle = \sqrt{3} \times 100 ( \sqrt{3} - 1 )$$\displaystyle = 100 ( 3 - \sqrt{3} ) \mathrm{(} m )$ [다른 풀이] $\displaystyle {\overline{\mathrm{CH}}} = k$라 하면 $\displaystyle {\overline{\mathrm{AH}}}$$\displaystyle \mathrm{=} \tan 60 ^\circ \times {\overline{\mathrm{CH} \mathit{}}}$$\displaystyle = \sqrt{3} k$ $\displaystyle {\overline{\mathrm{BH}}} = {\overline{\mathrm{AH}}} = \sqrt{3} k$ $\triangle {\mathrm{A} BC}$$= \triangle {\mathrm{A} BH} + \triangle {\mathrm{A} HC}$에서 $\displaystyle \frac{1}{2} \times 200 \times \sqrt{3} k$$\displaystyle = \frac{1}{2} \times \sqrt{3} k \times \sqrt{3} k + \frac{1}{2} \times \sqrt{3} k \times k$ $\displaystyle 200 \sqrt{3} k$$\displaystyle = 3 k ^{2} + \sqrt{3} k ^{2}$ $\displaystyle 200 \sqrt{3} k$$\displaystyle = \sqrt{3} \left( \sqrt{3} k ^{2} + k ^{2} \right)$ $\displaystyle 200 k = ( \sqrt{3} + 1 ) k ^{2}$ $k > 0$이므로 $k$$\displaystyle = \frac{200}{\sqrt{3} + 1}$$\displaystyle = 100 ( \sqrt{3} - 1 )$ 따라서 구하는 선분 $\mathrm{A} H$의 길이는 $\displaystyle {\overline{\mathrm{AH}}}$$\displaystyle = \sqrt{3} k$$\displaystyle = \sqrt{3} \times 100 ( \sqrt{3} - 1 )$$\displaystyle = 100 ( 3 - \sqrt{3} ) \mathrm{(} m )$