비례식 보기 판정

[출제의도] 유리식의 비 추론하기 ㄱ. $\displaystyle ad - bc = 0 \leftrightarrow ad = bc \leftrightarrow \frac{a}{c} = \frac{b}{d}$ (참) ㄴ. $\displaystyle \frac{a}{c} = \frac{b}{d}$에서 $a = ck , b = dk$ ($k$는 비례상수) $\displaystyle \frac{a ^{2} + b ^{2}}{c ^{2} + d ^{2}} = \frac{( ck ) ^{2} + ( dk ) ^{2}}{c ^{2} + d ^{2}} = \frac{k ^{2} \left( c ^{2} + d ^{2} \right)}{c ^{2} + d ^{2}} = k ^{2}$ $\displaystyle \frac{\left( a + b \right) ^{2}}{\left( c + d \right) ^{2}} = \frac{\left( ck + dk \right) ^{2}}{\left( c + d \right) ^{2}} = \frac{k ^{2} \left( c + d \right) ^{2}}{\left( c + d \right) ^{2}} = k ^{2}$ (참) ㄷ. $\displaystyle \frac{a ^{n} + b ^{n}}{c ^{n} + d ^{n}} = \frac{( ck ) ^{n} + ( dk ) ^{n}}{c ^{n} + d ^{n}} = \frac{k ^{n} \left( c ^{n} + d ^{n} \right)}{c ^{n} + d ^{n}} = k ^{n}$ $\displaystyle \frac{\left( a + b \right) ^{n}}{\left( c + d \right) ^{n}} = \frac{\left( ck + dk \right) ^{n}}{\left( c + d \right) ^{n}} = \frac{k ^{n} \left( c + d \right) ^{n}}{\left( c + d \right) ^{n}} = k ^{n}$ (참)