Anonymous
Anonymous asked in 科學數學 · 2 decades ago

有關面積的問題

△ABC,D、E、F分別在AB、BC、AC上,若AD:BD=1:3、BE:EC=2:5、CF:FA=2:3,求△DEF面積:△ABC面積=?

2 Answers

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  • 2 decades ago
    Favorite Answer

    令△ABC = 1

    連接AE,則△ABE:△ACE = 2:5→△ABE = 2/7 △ACE = 5/7

    又△ADE:△BDE = 1:3→△ADE = 2/7 × 1/4 = 1/14

    又△AEF:△CEF = 3:2→△AEF = 5/7 × 3/5 = 3/7

    連接BF,則△ABF:△BCF = 3:2→△ABF = 3/5

    又△ADF:△BDF = 1:3→△ADF = 3/5 × 1/4 = 3/20

    所求△DEF = △ADE + △AEF - △ADF = 1/14 + 3/7 - 3/20 = 7/20

    故△DEF面積:△ABC面積 = 7/20:1 = 7:20

    Source(s): myself
  • 2 decades ago

    設ABC面積為1

    由於斜邊比例等於高的比例

    對三角形BDE而言,底是ABC的2/7,高是ABC的3/4

    所以面積為(2/7)*(3/4)=3/14

    同理EFC為(5/7)*(2/5)=2/7

    ADF為(3/5)*(1/4)=3/20

    DEF面積為(140-30-60-21)/140

    29/140

    所以

    △DEF面積:△ABC面積=29:140

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